There is no Constitutional right to a secret ballot.
In the State of Oregon, all voting is done by absentee ballot. There's no privacy screen around you as you cast your vote. Your employer can stop by and say, "I'll pay you $1000 for your unused ballot, so I can fill it out how I want and submit it." If you're in an abusive family, your domineering alcoholic bipolar parent might force you to fill out the absentee ballot in front of them so they can control how you vote. There is no way the absentee ballot is considered a secret ballot, and yet we have no trouble when an entire state converts to voting by absentee ballot.
The State of West Virginia guarantees, in its state constitution, every resident's right to cast a public ballot. There's no mention of the secret ballot.
The secret ballot wasn't in use anywhere in the United States until it was first adopted by the city of Louisville, Kentucky, in 1888. The State of Massachusetts followed soon after. The first President to be elected by secret ballot was Grover Cleveland, in 1892.
We didn't use secret ballots to elect Washington, Jefferson, Jackson or Lincoln.
So, yeah. Anyone who claims we have a constitutional right to a secret ballot has an uphill road to hoe. History clearly shows that at no point in our nation's history has any court held the secret ballot to be a right.