I am just about to merge osTicket with this (free registration form).
All the functions, calls, pages are fairly easy to identify and it's simply laid out so easy to add/include your own css...
it does not seem a too difficult task!!
The only real php changes required for a full merger, other than the copy'n'paste of html inputs to your own pages etc. would be the merging of the database stuff, queries, syntax, defined names etc. ~ don't be put off by that though, if you can read and understand the structure of php database queries it'll be a doddle!
It's already setup to have a public area and a customer/members area, so it should be possible to operate support tickets (osTicket) from the members area and therefore exclude non-registered users from using the osTicket system.
I plan on using $_SESSION's to pre-populate the support ticket, so the user does not have to enter their details again after logging in. It seems the quickest and easiest route to get the merger working nicely without having to write extra DB code.
Have a look-see, hope this helps in the interim.