Loosing coolant and dumping transmission fluid?
I know that older LSs had a radiator that had separate compartments for coolant and transmission fluid, and that it was important when they were 10 or 15 years old to replace the radiator in case corrosion let coolant under pressure push into and mix with the transmission fluid.
If that's happened here, it's likely that the transmission made simply need replacing. And it won't be good for the engine running with insufficient coolant.
Myself, I'd get a different person to look, check the fluid, so when it came to it, it wasn't just your word for it, and maybe make a video showing the lost coolant/fluid.
Then I'd just take the car back and give them the keys.
Sorry.