I had a very similar problem but with a different TT. Here's how to find the culprit... Since the repair shop filled the grey tank and there were no leaks lets rule that out for now. Using only ONE SINK AT A TIME, fill the basin with soapy water and drain. Do it again and again with the same sink and keep checking for the leak. When you're convinced you've done this enough and that it must be another sink, then move on the next one. Do the same thing & keep checking for soapy water on the ground. Eventually you'll discover which sink is causing the problem - stop using that sink until you can trace the drain plumbing which might have a loose fitting or may even have a cracked pipe (thats what I found on mine).
As others here have mentioned you really should remove the underbelly cover to allow the bottom of the TT to dry thoroughly. A lot of work but you'll also be learning more about your TT.
2014 Micro Lite 19fd
2015 F-150 5.0L V8 XLT Crew Cab, 4x4, Tow Package, 36 gal tank, 3.55 locker, 1891 payload, Integrated Brake Controller, Roadmaster Active Suspension
Wooden Spoon Survivor