his title means less than that of the president-elect.
Uhm, no it doesn’t. The president-elect is powerless.
For a case study, look up the transition between Herbert Hoover and FDR.
In technical terms, you're correct. But seriously ... how many military folks are going to arm up on Dec 15 and go to war to enforce martial law declared by a president who has his bags packed and his china in boxes by the door? Especially if that president is thoroughly hated and has a reputation for acting unconstitutionally. And the president-elect is a staunch defender of America and its constitutional principles.
I would see a military refusing to obey the martial law directive and either waiting until the inauguration or peacefully deposing the sitting president and either installing an interim executive until the inauguration can be held or, as an emergency measure, inaugurating the president-elect early.
There is legal authority, and then there's PRACTICAL authority.