There is a process called reconciliation that gets triggered in the senate in a budget impasse situation. I would say it typically involves a whole lot of horse trading but at the end is a bill that can pass with a simple majority. So I don't think either party will be able to drag this out indefinitely or just to the midterms.
Dragging this out generally is a bad idea for Democrats. The cult following of stable genius is going to accept the negative consequences longer as long as they get fed a narrative along the lines of "we prevent immigrants from getting health care and drain the swamp of lizard people." Or whatever. The Democrats will feel their feeble support dwindle when unpaid government workers are done with their savings, which will happen before the cult runs out of patience. The whole battle is mainly fought on the backs of people whose only fault was choosing a career in government. The Democrats will take pity on them and eventually agree to the least dehumanizing compromise you can negotiate at the "12th" hour.
I don't doubt that. My crystal ball tells me they will compromise at some point for the good of the workers, for the good of the country - whichever line works best at that moment. I wouldn't want them to do that but I'm pretty sure it will pan out that way eventually.