LOL, I must have overlooked it as well. This is great. Thank you once more, @Manuel.
Anyway, this doesn't completely invalidate all the previously done research, since at least we now know how to make the hat work as both the joystick and the cursor keys. 
Did this work even better? How did you use it in practice?
I made a small program:
10 CLS 20 LOCATE 0,0:PRINT STICK(0),STICK(1) 30 GOTO 20
Then I ran the program and tried to move the d-cross in many different directions. It immediately jumped to my attention that STICK(0), which shows the state of the directional keys, and STICK(1), which shows the state of the first joystick, didn't show the same values as expected. Instead, STICK(0) seemed to lag a bit and depending of what I did, was just outright wrong. I think not all interactions were correctly predicted.
For example, for hat0 down, I think we should release left, up and right keys, not just up. At this point I was honestly a bit tired of trying more configurations, especially considering the hat0 binding to the joystick already worked better.
Anecdotally, I noticed I played Zanac EX much better with the latter configuration, despite still sucking at it.