Well it is clearly a constitutional issue. A good President would site constitutional-based rationales for extinguishing the DOE - in essence not executing Congressional law if there is one, regarding the DOE. This would hold true for all unconstitutional cabinet departments and bureaucracies which are the majority of the cabinet.
I’m not sure Trump has to have Congressional approval to nuke the DOE. But if he does, if Trump argued sound constitutional reasoning for extinguishing the DOE, he’d be OK and everyone would be better off.
Heck Obama has regularly ignored Congress and has remained unscathed - and his defiance was generally unconstitutional. Here, Trump’s defiance, if that’s what it would take to remove the DOE, would be constitutional,
Sorry...I used DOE(Energy) and should have used DoED
Dept. of ED was created by an act of congress. Not sure the president can do anything alone to adios it.