I had exactly the same problem, several colleagues also (JMP Pro 16.1 on Win10),
it was solved by @pmroz 's proposal, to run as admin, then you need to import that license file, and then it's installed permanently.
I had contact to Tech support on that also, they confirmed, that license is installed in the JMP installation folder. Then it's clear, that license installation w/o admin rights cannot be persistent due to missing write permission.
I think, when installing the license as std user, JMP should throw a warning, that this will not be persistent - but don't know, whether it will be considered to change (but I gave that feedback to Customer Support DACH).
Georg