Theoretically the LEDs cannot be affecting the RCD on a different supply as the RCD is only monitoring the outgoing circuits.
Possibly, but I am not sure how, the LED dimmer is introducing a DC component that is masking the RCD operation; alternatively if there is a Neutral Earth fault on the sunbed side of such resistance that is just tripping the RCD and there is also a neutral earth fault on the LEDs once switched on then this would give parallel paths for the neutral current and so reduce the current seen by the RCD below its tripping threshold.
If the sunbeds are switched on by a double pole switch then it is possible that there is a difference in arcing across the poles so that the RCD occasionally thinks there is an imbalance and the LEDs just happen to coincide with the trip times. This could be tested by connecting the "working" sunbed to one of the other sockets.
Possibly, but I am not sure how, the LED dimmer is introducing a DC component that is masking the RCD operation; alternatively if there is a Neutral Earth fault on the sunbed side of such resistance that is just tripping the RCD and there is also a neutral earth fault on the LEDs once switched on then this would give parallel paths for the neutral current and so reduce the current seen by the RCD below its tripping threshold.
If the sunbeds are switched on by a double pole switch then it is possible that there is a difference in arcing across the poles so that the RCD occasionally thinks there is an imbalance and the LEDs just happen to coincide with the trip times. This could be tested by connecting the "working" sunbed to one of the other sockets.