You are legally compelled by the government to fill out tax paperwork every year. Being (much more lightly) compelled to fill out a ballot is no different.
https://en.wikipedia.org/wiki/Compelled_speech
EDIT: IANAL, please do not cite this as any legal justification in a court if you fail to file any form with a government.
If your income is subject to taxation, you can't refuse to fill out the paperwork on the grounds that doing so lends legitimacy to an institution you don't support. The same should be the case for voting.
Mandatory voting would have the same requirements. Anybody who previously could vote now must.
In terms of state power: it's not all that great of an expansion. If you make over the taxable limit, you pay federal taxes. If you own any sort of land, there's a good chance you pay local taxes on it. These sorts of compulsions apply to (equally?) large swathes of the population without necessitating a great deal of state power.
The closest thing to compare compelling people to vote would be the draft.
Like I said above, a mandatory vote does not require individuals to make a political statement any more than taxation does, unless it is coupled with a limited selection of candidates. We can talk about the propriety of a mandatory vote, but the legality of one in the United States seems fairly cut-and-dry.
What you want is to make it easy to vote, and encourage it. Remove registration requirements. Make sure it never takes more than 15 minutes. Make sure it’s easy to know/find/access the polling place. Make elections be on holidays or sundays.
Do you know what ruling supports that? https://www.law.cornell.edu/constitution/first_amendment
And this article seems to support the idea that it would be feasible, and specifically talks to the First Amendment issue: https://www.theatlantic.com/politics/archive/2015/11/a-feasi...
You could probably write the law in a way that avoids first amendment implications. For instance, require that people show up at a polling place, but do not require them to submit a ballot.
If write-ins were removed, I'd agree that it violates free speech.
People voted for Harambe.
To me it is definitely a political statement to abstain from voting or abstaining from the political process entirely so its only logical that participating is also a political statement.