Anyway, such a system as you want has been designed, and is inexpensive and works well with existing systems, and makes it easy for third parties to audit the results: https://en.wikipedia.org/wiki/Scantegrity
However, a Merkle tree may actually be extremely helpful when tallying the results of various districts. Why not have each person simply receive a random off-the-record perfect forward secrecy token and vote once and have all the votes stored in a Merkle tree?
This system obviously requires centralized trust (which I’m not against in practice) — but is there any provably secure way to make such a system decentralized? (Yes, blockchain..)
In the Scantegrity sysem you can personally verify who your vote was counted for.