They put themselves in this position by voting down the bi-partisan commission in late May.
The commission proposal was designed to allow the repubs to name their own members, and the dems to name their own members, in equal number. (5 dems, 5 repubs, hand picked by each side).
They didn't want to do it because they knew it would demonstrate their complicity in the riot, and at that time the republican supporters were halfway between the "it's antifa making us look bad" and the "ok so it wasn't antifa, but it wasn't a big deal, there were no guns etc." phases of looking back on the riot.
So they felt they would get away with avoiding an investigation (and being complicit) without backlash from supporters.
As a result, the dems started their own commission, and have still asked for repubs to be part of it.
But it's theirs, rather than bi-partisan by design.
Unfortunately for the repubs, the supporters have now moved onto the "it was the FBI entrapping otherwise good and patriotic trump supporters. We need to find out who shot ashli and just how corrupt the FBI was seeking to make trump look bad and eventually bar him from future office. It was all a long con!"
So now, as a virtue signal, repubs have to pretend they want a commission and they are playing victim as usual with regard to who gets to choose members of the commission (even though they have caused this to be the case by voting down the original commission in May) as a pre-text to calling it all a sham so they can avoid complicity again.
Of course, the delightful thing about the current "It was the FBI!" narrative that is pushing the repub supporters to call for the repubs to do this commission, is that now all the supporters are primed to say "repubs and dems are all in on it, they were all complicit in stealing the election from trump and then staging the riot to prevent trump running again, so this commission is bullshit."
so that's both nice and awkward for repubs.