This sounds like it's the application's license model that, tbh, kind of sucks. Does the application vendor offer a concurrency based model that the licenses are pulled at run-time rather than requiring user intervention?
Is this a single key/code that needs to be entered, or is it unique to each user? If it's a single key, is there a way you could pre-load this in the image by either A) Embedding it in the software itself (Machine-based), or B) Adding it to the default profile (if it's user-based). If it's unique per-user could you use a product like UEM, FSlogix, Persona, or something to handle tracking the license entry so the user only has to enter the key once?