Wasn't there some previous discussion about there not needing to be a government shutdown, even in absence of a budget? I forget the specifics, but the gist was that the president could allow government employees to remain at work and performing their functions until a budget could be passed. Apparently this was the standard practice throughout history until some President used the move as a way of creating publicity and negative press against the Congress.
Assuming this is true, and Trump need not shut down the government, I think this would be the perfect outcome for him. Federal employees who are worth a damn or actually devoted to their country will likely remain at work. The vast majority of them don't need to be there for the cogs to turn and important functions to be fulfilled and it makes it so much easier to get Congress to cut later when you can point to the fact that things were running fine despite the supposed problems it would create.
Please Democrats, don't throw President Trump into the briar patch.