What if the Lapua Movement never fell out favor, by never attempting the Mantsala Rebellion? Could they have gained power in Helsinki? It would all hinge on Mannerheims support. But he wasn't necessarily anti-Lapua. And the Lapuans would most likely be more friendly towards the Axis.
You'd need an earlier POD than avoiding the Mäntsälä rebellion.
Generally speaking, to make Finland actually Fascist the Lapuans were too law-abiding. It was essentially a rural, conservative movement, and despite the rhetoric of some of its leaders too big a part of its supporters were not ready to go against the lawful authority of the first republic. See the Peasants' March of 1930 for example: even though it was a big show of force, it was nowhere near to a coup attempt and most of the participants in fact were very reverent towards the institutions of the republic, the president, the military leadership and the army.
By 1932 the movement was already seen as discredited even among the its own base, because of the kidnappings of Social Democrat MP and parliament vice chairman Väinö Hakkila and especially the respected former president K.J. Ståhlberg.
Also, by 1932 most of the movement's demands against the far left had already been fulfilled, for example the banning of Communist parties right after the 1930 elections. After this its support was on the way down and the Mäntsälä rebellion was an act of desperation on part of the extremist Lapua activists who were realizing that their historical moment had passed.
It is a paradox, really: to have a Lapua movement likely to make Finland Fascist you would need its rank and file seriously more extreme and anti-democratic, ready to destroy law and order to reach its goals. But a truly revolutionary right wing movement in post-late 20s Finland would never muster the supporter base big enough to overthrow the republic.
For Fascist Finland (or something so close to be undistinguishable from one), I would recommend a POD or PODs from the beginning of WWI to the early 20s. Making the Civil War or its aftermath go (even more) horribly wrong is my favourite starting point.
And Mannerheim? He was a career soldier, an anti-Bolshevik, an aristocrat and a royalist. He never did love democracy, but as traditionalist he believed in stability and duly constituted authority. He
did refuse the Lapua Movement's suggestion to become their candidate for president (/dictator). If 30s Finland becomes Fascist in the form of steretypical jackbooted thuggery, it is not a bygone conclusion that Mannerheim would even accept the command of Finland's armies come a WWII-analogue. The old man might have better things to do than to go and work for idiots, his love for Finland notwithstanding.