One key thing to remember is that the Finns at the time very much believed that Stalin's armies would be coming back to finish the job in the near future. Everything in the USSR's actions after the Winter War in 1940 pointed this way - not only the invasion and annexation of the Baltics, but also the way the Soviet government continually made demands on Finland, regarding the Petsamo nickel mine, say, or the re-demilitarization of the Ålands (something not demanded in the peace negotiations), and the opposition towards planned Finno-Swedish defence cooperation. The Soviets also supported the Finnish Communists and the Finnish airspace was continually violated by Soviet aircraft. In the summer of 1940 the Soviets actually shot down a Finnish Ju-52 passenger plane, the
Kaleva, en route from Tallinn when Estonia was being taken over by the Soviets. That the accident was not an accident but Soviet military action was not told to the people but the Finnish government and military knew it. In August 1940 a war scare broke out as the Finns thought a new Soviet invasion was imminent.
I have the habit of pointing out that the period after the Winter War was called "Interim Peace" already during the time, and this was not because revanchist feelings but due to the fear for that new invasion. Sure, many people in the military and in the political right wanted to join hands with Germany for revanchist and irredentist reasons. But without that existential fear for a Soviet takeover, and the untenable place Finland was in in terms of trade and economy (Germany and the USSR together being in a position to stop virtually all Finnish foreign trade if they wanted to), the highest political and military leadership would not have consented to allying with the Nazis.
It is very difficult to see Finland staying neutral - in my view this would require the USSR dropping the idea of conquering Finland and being ready to allow Finnish defence cooperation with the Swedish, as well as being generally ready to treat Finland as a benevolent neutral instead of a hostile nation on the waiting list to be conquered. This would also mean selling Finland cheap food, etc, to alleviate a potential German de facto embargo on Finland.
Then, of course, the Finns would after the Winter War be very sceptical towards any and all Soviet moves anyway, be they hostile or friendly, and if the USSR was suddenly too friendly the Finnish government would probably see it as a cynical Soviet trap to steer clear from.
So while I don't see the Continuation War as the only option, I think neutrality would have had only a very, very small chance even after the official Soviet line would have been to foster Finnish neutrality and keeping it out of the war after 1940. The Germans were looking into making Finland an ally on the northern flank in an upcoming assault against the USSR since 1940, and due to geopolitical reasons they were the only major power Finland could really turn to for protection against Stalin. They had all the weapons of persuasion, bribery, blackmail and threats to wield against Finland and to turn the Finns heads to cooperation on some level.
It is IMHO pretty hard to go against those odds, unless the Finnish government would be ready to take the chance of staying tenaciously neutral in spite of this possibly leading to many of its citizens starving, the military not getting the weapons it sorely needed and then both the Soviets and the Germans making war on Finnish soil and the Finnish sea areas and airspace
anyway, with the Finnish state being at war against both.
