As soon as it looked like we were headed into another shutdown, my guess was that their play would be to argue, "alright, it's shutdown, I guess we don't have a government anymore. I have no choice then, but to make a new one."
Which you'd think would be a widely unpopular move, until he comes in with "and we'll just use the original US constitution, no extra stuff like 'amendments', just good ol' 1776 like the founding fathers intended".
I think his base would actually get behind it.