Your best bet for a (paid) gfe experience is with an Internet escort. I've posted about some of the agencies elsewhere. Or visit TJ Amgios. It's a free site with whole whole forums devoted to bar girls, escorts, etc.
As far as finding a civilian girl who wants to "shack up" while you're in town, there are no agencies I know about that arrange such things. Chicas in TJ are very used to rubing shoulders with American men; those looking for such relationships have lots to choose from. (It's next to the US; plus supposedly, 196,000 Americans live in Tijuana.)
Most TJ civilians chicas are "street wise." You'd probably have trouble establishing a relationship in a few days. (It's another story--I'm told--in some of the small towns in southern Mexico.)
You can, however, get some nice TLN (all night) sessions with many bar girls for around $250.00. (Especially during the week.) That might be the best route for what you want.