They could solve the House of Representatives problem with a single vote if they used ranked choice voting. Here's how it would work:

All members are given a list of all possible candidates.

Anyone willing to be considered is on the list. The Democrats could nominate some additional choices after Jeffries if they want. Both parties could nominate members of the opposite party they consider more acceptable than those that other party put forward itself.

All House members would have enough votes to rank all potential candidates in the order they would accept them.

Now it's a single ballot, but with all the candidates ranked in order of each voter's preference. The rounds of "voting" are all taken from those single ballots.

Person with fewest votes is eliminated after first round. Votes of people who listed him (and holy fuck they're STILL all hims?) would then go on to count for their second choice in the next round.

Next the person with the second fewest votes is eliminated and the next choices of those who chose him move up. And so on.

A member can decide not to rank any candidate they consider absolutely unacceptable, but if they run though their entire list of candidates and none of them is chosen, their blank line votes start counting as abstentions, so the 217 goal for choosing a winner drops lower.

It continues until someone reaches a majority of non-abstention votes.

Any member can end up casting a vote for a member of the other party without suffering political blowback, because after all, it could have been their very lowest ranked choice.