Have you tried:
1) After executing PHAP_ADMIN, and before you get to this screen.
2) Goto se24 (Display class cl_exithandler)
3) Double click on the method GET_INSTANCE.
4) Put a break point at Line no.25 (CASE sy-subrc).
Now
5) Press the required button on PHAP_ADMIN to get to your next screen
6) Check the values of variable 'exit_name', it will give you the BADI name called at that time.
7) This way you will find most, but not all, the BADIs called
