Here’s the problem...If you have to sign the outside of the mail in...it’s easy enough to look up your party registration. and “take action”.
Don’t give an inch. They wouldn’t.
In Michigan for the primary we just had, you had to request a Dem or Republican ballot. They are also numbered differently starting with first digit. Example all dems start with a 3 and Repubs start with a 4. So folks know up front whether it’s republican or dem ballot.