It looks like neither side can back down. He will have to invoke the emergency and designate the funding and then he can sign whatever they put in front of him. That will enable them to say they held tough for their base and hopefully (for them) give them an issue to run on in 20.
If he invokes an emergency order Dems will oppose it in court. He will then refuse to sign a bill to reopen the gov thereby proving the Democrats oppose securing the border. Or something like that.