My GF is the same way, has to be well done or past, although she is starting to come around some to a slightly less done piece of meat since I showed her it is not as tough if not over cooked. But when it comes to cow, ground up and an occasional strip steak.
If cooking the tritip to her liking takes it to far past where it is good, maybe cook to your preference and then cut some off and finish it off in a pan toget it to her liking.
For me give one of those bottom pieces of meat a sharp knife and some salt and pepper and I would be good. Of course I would want it closer to room temp than just out of the refrigerator.