Well, I've gotten half a decade with the assumption that they had a King, so...
The offer came with a bit extra added, namely Kola and Karelen*. Could that swing things more in a pro-Royal and pro-Swedish direction?
*And also swearing off all future claims to the Swedish throne, and promising to protect the rights of all Finnish-speaking.
All right then. Here is a take of a possible timeline. It is admittedly a stretch, contains a fair bit of improbabilities and assumes that the royalists would succeed in everything they try. Also, I am not sure Prince Carl had the personality required for this...
Anyway, comments are welcome.
How the Kingdom of Finland Got Its King: A Timeline
March-April 1916: Tsar Nicholas II of Russia, the Grand Duke of Finland is deposed
April 3rd: By orders of the Provisional government, the Russian leaders of the Finnish Senate are arrested and sent to Petrograd
April 10th: The Provisional Government reaffirms Finnish autonomy
April 15th: The broad-based Tokoi senate appointed in Finland
April-May: The Parliament considers the question of highest power in Finland.
ITTL, the Swedish, together with the SFP and other conservatives have been conducting a PR campaign for Sweden and Finnish independence since 1915. By this time, most non-socialist parties have tilted a lot towards going for independence with Swedish support, when as the Social Democrats are divided and many of them maintain the opinion that the question should be put to the Provisional Government in Petrograd first before doing anything
rash.
May 20th: The conservatives present "Lex Tulenheimo", the Power Act. It is accepted by 115 votes to 79, with the SDP group splitting half in the vote. The Parliament declares itself as the highest power in Finland; the SDP wants to have the Act affirmed in Petrograd.
May 31st: Without waiting the answer from Petrograd, conservative senators led by Tulenheimo present the Independence Act, which is accepted by an even smaller margin, 101 to 91. Social Democrats vote against it, and the Social Democratic senators walk out of the senate.
The SDP still considers the Provisional Government legitimate and see the move as unconstitutional, when as the conservative side has been radicalised by the Swedish (and by this time, German) goading and promises. The SDP would not be against eventual independence as such, mind you, but ITTL the events are moving too fast for them, and not being privy do to secret deals possibly made with the Central Powers, they do not grasp the full implications of the situation.
June 2nd: The conservative Setälä senate formed. As the first order of business, it declares Finland independent.
June: Russian units in Finland grow increasingly demoralized are in a process of breaking apart. Some are forming revolutionary councils, others are using their own initiative and withdrawing quietly towards Petrograd. Both the right and the left are forming militia units, at this point really as an effort to maintain order during the elections due in early July. Even so, both get into scuffles with each other and the Russians, even if it is nothing major yet.
The electoral campaign for the elections is started. The conservatives and the SFP use the proposed peace treaty as their electoral platform, advocating a strong Finland "from Hanko to Petsamo" in alliance with Sweden and Germany. The acceptance of a Swedish king is presented as a guarantee of security and a strong rule of law. Horror stories of the situation in Petrograd and the chaos in Russia are circulated as evidence that "the way of the SDP is the road to anarchy".
Where Swedish troops cross the border to Finland, they are treated as liberators. The conservatives ask the Swedish to make a landfall near Helsinki, but the Swedish command refuses because of the unclear status of the remaining Russian units at Krepost Sveaborg.
The conservative campaign features prominently several declarations ostensibly written by the official candidate for king, Prince Carl of Sweden. The declarations, beginning with "Long live Free and Independent Finland!" stress that the Prince will only accept the throne if the Finnish people so will, and that if he comes to Finland, he swears to protect the rights of the Finnish people, honor the legitimacy of the Parliament and "not to rule the Finns, but to rule with the Finns".
Both capital papers, the Hufvudstadsbladet and Helsingin Sanomat (and many provincial papers beside) are fully mobilized behind the royalist campaign. The SDP is painted as irresponsible, Russophile and revolutionary. Much is made about supposed coöperation between the socialist extremists and the Bolsheviks in Russia (of which at least some part is true).
Later historians will consider the conservative campaign wildly successful.
July 5th: The official results of the 1916 Parliamentary elections:
SDP 75 (-15)
Finnish Party 43 (+4)
Young Finns 32 (+3)
SFP 27 (+2)
Agrarians 24 (+6)
The results see a clear shift to the right. Comparatively, the biggest winner is the Agrarian League, which has benefited from the move wihin the moderate left to the centre. For the SDP, the result is catastrophic, their worst during the reformed Parliament. The turnout is exceptionally high (76,8%), and the increase appears to have benefited all the other parties but the SDP.
July 12th: A conservative-liberal senate is formed, led by Svinhufvud (Young Finns).
July: The tension between the Red and White militias nearly reaches crisis point. The left fringe of the SDP accuses the conservatives of election fraud, but can not prove the allegations. The Russian Fleet withdraws from Helsinki, and the different militias fall over themselves to hoard weapons left behind by the Russians.
Transport ships of the Swedish fleet arrive in Helsinki and bring along infantry units "to help the government to maintain the peace". The move is condemned by the left, while the conservatives have parties in celebration of the arrival of the Swedish army.
While the Parliament discusses land reform, demanded by both SDP and the Agrarians, the royalists led by Svinhufvud demand an election for a king, citing "an empty throne" as per the 1772 Instrument of Government.
The pieces are carefully set: behind the scenes, the Agrarians have been promised, "free hands" (within reason) in the land reform issue if they support the motion, whereas the liberals have been persuaded to join in support by promising them a strong say in drafting the new constitution. In the end, on August 5th, the Parliament decides to elect a king by 101-91, with an extremely narrow margin because in the event both the Young Finns and the AL show internal fractures.
On August 6th, Prince Carl of Sweden is unanimously elected the king of Finland by 101 votes, with the SDP and republican agrarians and liberals boycotting the vote. Because the 5/6 majority needed to make the decision into law is during the present term is not reached, the decision is left to rest over the next elections.
August 8th: The constitutional committee, led by Ståhlberg (Young Finn), begins to draft a proposal for a constitutional monarchy. (Which will favour the Parliament, IOTL Ståhlberg was a leading republican.)
August 15th: The SDP and the Unions declare general strike. Large socialist rallies in several towns. In Tampere, a Red guards unit opens accidentally fire on passing members of the bourgeois militia: 9 die in the exchange of fire. The senate urges restraint on all sides and mulls over a proposal to declare martial law.
August 22nd: In a daring move, Prince Carl arrives in Helsinki with additional Swedish military units, to the consternation of his advisors who consider the situation at the Finnish capital too dangerous. The Prince is determined to make a show of his support for the Finnish government and to get acquainted with his future domain. His arrival is kept as a secret.
Unfortunately, the same day has been chosen by the revolutionary wing of the SDP as the beginning of the takeover of the country. Led by Eino Rahja, a close associate of Lenin, the group plans to take Helsinki and by proclaiming a People's Republic there to turn the General Strike into a bona fide revolution.
At 6 p.m., the signal is given and the revolutionary Red Guards in Helsinki take the streets. At Market Square, the unit tasked to take control of the area runs into a Swedish contingent moving from the port towards Senate Square. A order to fire is made. The military entourage of Prince Carl returns fire: the Swedes manage to take cover while the Reds break off towards Kruununhaka. One of the Swedes is dead, two are injured: the Prince has caught a bullet in his arm, but is otherwise unscathed. The wound receives a field dressing, and the entourage reaches Government Palace without further interruptions.
In the capital as in other town in southern Finland the militias clash for days, with Swedish units aiding the White side - declared as government army. The conservative press sings the praises of the Prince, who is lauded as a martial hero. The Reds are accused of an attempt to "assassinate the future king of Finland" and condemned to the lowest levels of Hell. The Prince receives a huge boost of popularity, especially after he decides to act all modest and downplay his role in the fight against the revolutionaries.
In the end, the moderate SDP leadership denounces the insurrection and most socialist will have nothing to do with it. The revolution fizzles out by September 3rd, when the last Reds surrender at Hämeenlinna. About 300 people have lost their lives and a 600 or so injured: in comparison with OTL Civil War this insurrection is very limited. "Supreme Commander" Rahja himself is found dead, and most of the Red leaders arrested. Some of them are executed for treason, others go to prison.
In Helsinki, a victory parade is held: in effect, it becomes a celebration of the Prince. Carl himself decides to stay in Helsinki to await for the time his role is confirmed; he is also quite excited of the situation he finds himself in.
October 1st: The royalists choose to ride the wave of popularity. Svinhufvud, as Regent, uses his powers to dissolve the Parliament. He calls new elections to be held in February 1917. Svinhufvud cites the dissatisfaction of the working class and the doubts about the fairness of the previous elections as the main reasons behind the move, but generally most people see that the greatest reason for new elections is the need to clarify the situation with the governance of the country.
October-January: The electoral campaigns mirror those of the previous year, but now both sides aim for a more conciliatory tone. The conservatives call for a movement to unify the country under one strong leader: the role of the Prince in the recent events is reiterated time and time again. Prince Carl himself goes on tour of the regional centres, providing much good publicity for the royalist cause. The SDP campaign is muted: most importantly, the party majority tries to distance itself from the Red insurrection and professes its support for peaceful political development.
February 6th, 1917: The official results of the 1917 Parliamentary elections:
SDP 73 (-2)
Finnish Party 43 (0)
Young Finns 31 (-1)
SFP 27 (0)
Agrarians 26 (+3)
The Agrarians are again winners, but by and large the election confirms the previous year's result. Conservative-liberal senate is formed, led by Setälä (Finnish Party).
February 20th: The Parliament confirms Prince Carl as the King of Finland by 121 votes to 59. Some of the Social Democrats boycott the vote, but mostly because they dare not to vote against it due to the immense popularity of the Prince and the possible crisis that his rejection by the Parliament would cause.
March 15th, 1917: The coronation of
Charles the First, King of Finland and Karelia, Grand Duke of Lapland, Lord of Kaleva and the North.
(The list of titles might be just a myth, but it is fun even so..)