Transponders and ATC's Radar

That sounds about like what my avionics shop said. I clearly remember
them saying something about the transponder being slightly out of phase.

I have heard about problems where the transponder appeared inop at one
class D, but was good at the next a few miles away. Apparently the
transponder's frequency was slightly off and some ground equipment
could read it, while the other ignored it.
That there should be an issue simply because a different controller has
you on his screen sounds odd.
As others have suggested recalibrating your transponder (or replacing
worn out tubes) should make sure that it operates on exactly the right
frequency.