Bearing in mind the coalition governments that the OTL Soviets initially organised in various East European countries after WW2, those "key positions" would probably be the ministerial positions controlling the armed forces and police... thus (they would hope) making a complete takeover slightly later on somewhat easier.
IOTL the Soviets put pressure on the Finns on which people and parties (the National Coalition) would be "undesirable" in the cabinet and that was tacitly accepted. The Soviets also demanded that the SKDL, the new far left party be taken into a coalition cabinet and that happened too. In 1945-46 (Paasikivi's third cabinet) the SKDL had Interior and Defence, in 1946-48 (Pekkala's cabinet) Prime Minister, Interior, Defence and Justice.
So, even without demanding key cabinet posts explicitly in 1944, the Soviets managed to get "their" men into those positions by 1946. That was IOTL. They also took over the State Police (Valpo), and during the Pekkala cabinet it was known as the Red Valpo, acting against the right and the war-time establishment.
There was also the shakeup of the military that resulted from demobilisation, the banning of "Fascist" organisations like the Civil Guards and of course the Weapons Cache Case in 1945-50. The Finnish Defence Forces never had any new Communist officers though: just those deemed, again, "undesirable" by the Soviets were booted out and "uncompromised" officers put into leadership positions.
And what with the heavy reparations and the strong position of the Allied (read Soviet) Control Commission in Helsinki, the Finnish economy was also pretty curtailed in the immediate post-war years.
What it comes down to is that the Soviets pretty much got what is outlined in the OP (apart from more military bases and Communist officers), not through direct peace terms but just putting pressure on the Finnish government and parliament in 1944-48. This is to me ironical, because if what was reality in, say, 1947, would have been demanded from the peace delegation in 1944, they might have even walked away. Because there was no actual occupation, Stalin seemed to have decided on a gradual takeover using a truly local "vanguard"(ie. the finnish Communists reorganised as the SKDL) as a weapon of choice.
And still Finland didn't go Red during what we call the Years of Danger. The thing is, even if the Finnish left was given a lot of backup by Moscow, they still were not strong or brazen enough to try to step up and take over in earnest. They simply were too weak to act on their own - and Stalin would have wanted the initiative be Finnish, the revolution to really start in Finland before the Red Army would interfere in support of the new popular government.
Falecius said:
The US probably wouldn't care about Finland enough. They'd just let it fall into the Soviet sphere. They were perfectly happy with Finland as a military "ally" of the Soviets IOTL, so I doubt they would interfere much TTL or change their policy elsewhere in significant ways at the beginning.
That was another of the little ironies of the situation. The US might have accepted Finland going Red, but Stalin was actually wary of a potential US reaction to a Soviet takeover in Finland. The US had been very positive towards Finland in the 30s and through the war even when it was a German ally. It really seems like what Stalin thought the US might do prevented at least part his taking over Finland in 1944-48 - as that was one reason behind wanting a truly Finnish revolution/coup to begin a possible post-war occupation.
This is something I think many on the forum might not accept, considering Stalin's other bold moves in the time period in question. But IOTL the Soviet Union could have occupied Finland in 1944-48 and incorporated the nation into the fledgling Warsaw Pact *like that*.
A few tank columns moved from Porkkala to Helsinki, taking over key positions and the Finnish Broadcasting Company. The Red Valpo corraling up leading anti-Soviet politicians and army officers. A few divisions crossing the border to beat the already-gutted and leaderless Finnish army in Southern Finland. A statement by Prime Minister Pekkala about the Soviet troops coming in to help put down an illegal Fascist/army coup. Special Soviet-trained Finnish-speaking militia units taking over where the ordinary police has been ordered to stand down by the Communist Minister of Interior. Some sporadic fighting by small army units, some of the navy escaping to be interned in Sweden. Estonian-style "forest brothers" fighting a losing guerilla war against the Soviet occupier in the forests of the north. By the early 50s Finland would have been firmly in the Soviet camp.
But this didn't happen, because Stalin though subduing Finland by force would be
more trouble than it was worth, in terms of foreign politics and militarily. Now that Finland had already been "pacified" and was not a threat to the interests of the USSR anymore. Unless, of course, the Finnish left would do most of the fighting for him. But, like said, it was too weak for that.
A Finnish People's Republic in 1944? I think a thorough Finnish defeat in the Battle of Tali-Ihantala in the summer and a subsequent Red Army drive to Helsinki is still the most plausible option.