I don’t know how this will turn out, and maybe what I’m about to say will be disproved by events, but here goes:
I think the Republicans face a difficulty in electing a Speaker that the Democrats wouldn’t have, and it will be hard to overcome. Democrats may disagree intensely, but they all have legislative agendas to pursue, and in the end they are likely to compromise in order to get at least some of what they want. Republicans have little to no agenda. In the last presidential election they didn’t even have a party platform. Thus there is no incentive to compromise. If you’re a Republican congressman eager to cement your brand as a “patriot” who won’t settle for RINO’s like Kevin McCarthy, what would motivate you to vote for him?
True, representatives, even very right wing ones, still want federal money for their districts and to win favors for friends and donors. But these things usually take the form of riders to bills for other purposes or fine print in legislative language. The whole point of the process is that it occurs out of public purview and is therefore difficult to use to break highly visible logjams like the speakership. The IRA compromise among the Dems did involve side payments to West Virginia but primarily took the form of substantial trims to programs most Democratic senators supported.
What will a compromise that assembles a working Republican majority in the House look like?