The dealer is going to charge you a MINIMUM of an hour rate. That’s normal.
You should NOT be paying for one dime of this. The quick lube place should be paying for 100% of the transmission replacement. Not only that I’d have them pay for a rental and any lost work time. They screwed up and should pay for it.
This is just another example of why to stay away from these quick lube places. They hire high-school drop-outs who don’t know enough or don’t care enough to do the work properly.
There should be a “Holms on Homes” equivalent show for cars.
I wouldn’t worry about the diagnostic costs. It would be better to have them go through the transmission and be sure of all problems rather than do a band aid repair and have the car back in a couple weeks for something they didn’t find on the first visit. I agree they can probably tell within 1-2 minutes the transmission isn’t right, but it may take longer to find the exact problem(s). Since someone else damaged the transmission you can also send the diagnostic bill to them/their insurance company along with bill for the repair.
Don’t worry about the kid’s job. He’ll get hired down the street and do it to someone else’s car. My BIL was in a time crunch and took his car into a quick lube. He just happened to be watching when the “kid” grabbed the tranmission fluid hose to replace his engine oil. Kid just kinda looked dumb and said “oh yeah that happens sometimes”.