The traditional case on the matter is
Youngstown Sheet and Tube. Jackson's opinion is the one that modern jurisprudence tends to follow.
The key issue here is that appropriations are strictly a congressional power, so unless Trump can point to a provision allowing him to use funds, he loses. He is probably going to point to the statute that I mentioned. You've keyed in to the first hurdle Trump is going to face, but there are others. Here's my list:
1.)
Is there an emergency?
2.) Does the emergency
require the use of armed services?
3.) Is the military construction
necessary for the use of armed services?
4.) Is this the sort of "military construction" that congress meant when they created this particular emergency power? (This is central to the Youngstown constitutional argument - if it isn't, Trump loses)
4.)(b) Is this affected by the budget requirement that the border security funds should not be used for the wall. (this is the issue Ann Coulter likely refers to when she says that Trump has to argue that the budget is unconstitutional)
5.) Are the funds unobligated? (probably)