I wish I could tell you. I can only give you some words of warning. Toyota will or won't do it even for a Toyota part. That you know. The install of a clutch, flywheel, and LSD from Toyota will be over or close to a $1000 for labor. I wouldn't trust them to add the LSD. The rest is highway robbery.
I was quoted from a shop to do all for around $450. I won't go back there and they aren't there anymore even if I was stupid enough to go back. You have to be careful on choosing a place that will do the install right and is reputable.
You are doing the right thing by asking and keep on asking everywhere you go and everyone you meet. I wish I could help because I'm in the same situation.
Where I live (and not Cali), to find a place that knows what they are doing for any sort of transmission, clutch, flywheel, most important LSD install/replacement is tough especially a place that never touched a 7th gen Celica, it's transmission, or even a part that was never there to begin with. Most places here can replace everything included in a 88 Dodge truck's transmission blindfolded. I eventually found a couple places regardless.
Search around for places that specialize in drivetrains or clutch replacement for racing or any place that does transmission work on higher caliper cars in your area (on most counts the labor cost will be less than Toyota and more trust worthy).
Believe it or not a little work on the computer can yield loads of info in your area even testimonials.