MosquitoXEL
From back in my days as CFI:
We had a nearby Class C airport with a Class D airport tucked under the C shelf kinda in its armpit to give you a visual.
DPE favorite especially with foreign students to do a touch and go at the Class D.
ATC controller would generally cancel radar service and tell you to squawk VFR when handing you off to the D TWR.
Which means you’d be better off doing a full stop taxi back and coordinate with TWR or GND freq for a transponder code outbound.
Otherwise you’d have to stay below 1200’ while contacting C getting a code and explaining which way you want to go while maintaining your 1000’ above the highest obstacle.
So the solution was:
1. Coordinate with C controller to keep the code for your outbound and just having to establish radio contact after release from D TWR, or
2. If for what ever reason you couldn’t make that work ( frequency congestion etc ) do a full stop taxi back to coordinate the outbound and tell the DPE that you can’t comply with his request for a touch and go.
I’ve even had Commercial applicants on stage checks bugger that one up.
Thats why your story kinda sounded familiar.
Best of luck.