Andrew,
I don't think a meeting will do the magic you're hoping for... but a Webinar would. You could put the non-voting participants in the Attendees, and the voting participants on the Panel. Of course, that's an extra expense, and not sure whether you need to do this monthly, or less often (or more often).
A better option would be to find a voting app. Some of the online voting services let you vote free for up to a certain number of voters. If I recall correctly, ElectionRunner (which I've used in several venues) is free for up to 20 voters.
I'm also associated with the National Association of Parliamentarians, and think you might want to discuss your voting needs with a parliamentarian in your area. Your organization might be over-thinking the voting requirements in some cases, and could be simplified (especially if you have a small board, like an HOA - which I also have experience with!).
Also... the pre-loading of Zoom users being sent to Breakout Rooms is dependent on their Zoom login email address - not their Profile Name while in the room. Yeah... it isn't easy, no matter how you slice it.
One more option: Use the Reaction feature, and assuming the voters are voting for or against something, have them use the Yes/No Reactions. If this interests you, let me know and I'll elaborate.
Good luck!!