The Democrats should nominate a sane Republican for Speaker of the House. I bet there are enough pragmatic Republicans and Democrats to get a sane Republican elected so that Congress can get a few non-controversial things – at least things that shouldn’t be controversial – done, e.g., passing spending bills, passing a transportation bill, raising the debt ceiling. Given how things are playing out within the GOP caucus I think that looks the most constructive path forward. Would enough Republicans go for it? Who knows, but I hope the Democrats are at least making inquiries.
The political consequences? I figure if that’s how things went down then the Democrats gain a few seats in Congress next year – although not a majority – and almost certainly win the presidency. The Republicans lose some seats but probably maintain a majority in the House and possibly the Senate. So what’s in it for them in compromising? They get to demonstrate that at least some of their caucus is willing and able to govern. It helps them rebuild their “brand” as a sane center-right party and it gives them a better shot at the presidency in 2020.
Oh, and you want to keep the number ideologues in Congress to a minimum? Have open primaries. The top two votegetters move on to the general. Sure, some of my preferred candidates wouldn’t make the cut but I think it would be worth the sacrifice if it would cull the goons from the herd.