Trinity as a first name is definitely klassy. I think stripper, then Matrix, then religion.
Fiona does have the Shrek connection, but I never think of it unless someone points it out. There is also Fiona Apple, Fiona from "Burn Notice" and there was a teen Fiona on an old Disney show called "So Weird". There's also a really annoying dog at my park called Fiona. Anyways, I think the name is out there enough that most people won't link it to anything in particular.
I do think Fiona Trinity flows well if you want to compromise.