OK, sure, he can shut down the government arguing with his own party, that is, assuming the GOP retains control of congress considering that they have done absolutely nothing to warrant such.
No, he does not have to shut down anything. Democrat Harry did not present a senate budget for 4 years. It did not shutdown the government. Please quit before you make more illogical arguments.
Again the main point is the President has a lot of power with the veto threat and can push the congress to bend his way, if he/she really wants to and has the courage to do so.