Yes the CAA does have jurisdiction and there has been precedence.
The FAA certificate is based on the underlying UK CAA certificate which means operating an N-reg still puts you under CAA authority.
They could do an emergency revocation which would make your FAA piggyback invalid.
The U.K. license would not read everything prohibited except the following but it does say Single Engine Piston but does not explicitly prohibit a SE turbine as it does not read SE Piston Only.
But these are the cases where it's just simpler to get an FAA stand alone certificate to avoid any legal interpretations down the road.