Enacting this rule only requires a simple majority, not a two-thirds. You'd still need two-thirds to remove, but the Treason Partiers have an "out" if the ballot is secret. Probably Moscow Mitch would have to introduce the rule change for it to be voted on, however, so short of a true revolt in the Treason Party, don't count on it happening.

A vote for a secret ballot rule would be vote to remove Trump. The vote on the rule would not be secret. It therefore seems unlikely.

It would be great but it seems unlikely.