eyeTV 4 is currently not a full release. The Beta version needs a different Key from your eyeTV 3.
Search eyeTV for:
Eyetv 4.x for Mac 64 - bit (public beta + preview)
This has 5 keys that will allow eyeTV 4 to work. The keys are temporary (until 30th November or whenever eyeTV is officially released.
No idea if your eyeTV key will then work or not.
I will be moving over to Mac OS Catalina soon and need to be sure that eyeTV 4 works before I change. I am already having to pay for Microsoft Office as my old version won't work. I don't want to change to Catalina only to find that any more of my programs are no longer working properly.