Trump already has authority and money to build his wall -- he just hasn't been reminded of it.

Curiously the Corps of Engineers could build the wall as a normal War Games exercise.
Scenario: enemy troops headed for position, their air attack capacity is neutralized but ground troops and vehicles a major threat.
Goal: build fortifications that would stop troops in foot and any vehicles that reach a key perimeter area.
Method: standard barbed wire on anti-Trump barricade, with metal personnel barrier that can stop conventional rifle or pistol gunfire.

There is a need to test structure, and it would have real application if the war in the Middle East were to spread past Turkey and into Europe where NATO would need to respond quickly.
Clearly, the southern USA border is an ideal test ground for America Troops to practice construction and, migrants would serve to provide real world test of efficiency against a determined but unarmed force.
The War Games budget and Engineer training Budgets would pay for it from allocated funds.

If the USA can engage in war games around the world, certainly there is nothing improper in engaging in related work within the USA, with an eye to transferring the developed skills to the future theater of operations where they might be needed.

No emergency declaration is needed.  Not goes Congress have authority to challenge peaceful military training.

Of course, Trump could, since it is an exercise in preparation for a NATO related threat, use funds earmarked for NATO.  He could even pulls troops out of Europe to train, and thundering their pay etc back into the American economy.
If NATO nations wish to complain, they can provide the  materials and test training areas necessary for the exercise to be conducted in Europe -- and they can enjoy the disruption to their open border economy as the military deals those borders with steel barriers the EU will need to provide locally as part of the exercise.
Or NATO can state clearly they do not sarcaStic associated with protecting their borders from land or sea attack.  And America will henceforth cease involvement in anything related to such a threat.  Effectively telling America it does not need it as part of NATO and Europe will protect itself from all threats internal or external.