The drain and fill is only like 3-5 quarts out of the 12+ (?) that is in the trans. You want fresh fluid to try and stop the slipping.
The stuff isnt cheap and there is a lot of it, so unless you are going to DIY, then have the whole thing flushed with fresh Toyota WS fluid.
I get them not wanting to touch it but maybe find another place that will, and youll have to know going in that the $300 or whatever is costs might be lost if it doesnt fix it.
If the flush doesnt fully solve it then a last ditch effort would be to add some 'Shudder Fix' stuff and pray lol.