Get rid of the GFCI breaker... Use a standard breaker instead.
The reason you don't want them (GFCI) is because the mobile connector will, on purpose, send current down the ground wire to make sure it is connected. GFCI breakers that are very sensitive will interpret that as a ground fault and trip. The Tesla Mobile Connector has built-in ground fault detection.
Regarding the dropping charging current, that's almost always a bad breaker (seeing a pattern here?), improperly torqued wires to the breaker and/or outlet, or bad outlet (heating). Does your car show anything in the service screen notifications when charging current drops?
Since you've swapped the UMC, that rules out the UMC as the problem. Simple process of elimination suggests it's either the car or the breaker, outlet, or wiring. You can plug the car in with the UMC and have no issues elsewhere, so it's not the car... That leaves breakers, wiring, and outlet. Since you've got GFCI breaker and replaced with another GFCI and have issues, replace that with a non-GFCI breaker first.
Then, get your electrician to tell you EXACTLY what gauge wire is installed.
But, assuming a competent electrician, the GFCI breaker is the likely root cause. But you need to have every single connection point on the circuit checked for correct torque.
Things you don't want:
GFCI breakers
Aluminum wires
Splices in the circuit that aren't Polaris type or equivalent (I'd not want any splices in the circuit at all)
Undersized wire
Lowes/Home Depot type consumer grade outlets
Incorrect torque on any wire connection
Bad ground
