What kind of bill, exactly, could the GOP propose that Democrats could support without abandoning their core principles?

You can't flip this around to 2008, either. Democrats ended up passing a bill that was fundamentally the same as a bill passed by a Republican governor and had, in its ancestry, a proposal from the Heritage Foundation. 

Republicans opposed the ACA because a) Democrats were proposing it and b) a large bunch of Republicans want the health system fully privatized.

Democrats are opposing the current GOP proposals because a) it will take health coverage away from millions of Americans, b) it will likely make health care cost a lot more for a lot of Americans and c) Republicans are proposing it.

There's no real equivalence here.