One thing is for certain, it won't have been any sort of overload or in-rush current that tripped the RCD as RCDs do not provide any sort of overload protection. That is why they are always used in conjunction with standard MCBs. I'd say that it was almost certainly some residual current escaping through the CPC that has caused this.
Combined Earth Fault and Overload protection is achieved using RCBOs as they are a one stop shop and don't need to be used in conjunction with MCBs.

Problems like this can be difficult to solve without the proper test equipment because you don't know if it is the device that is tripping the RCD on its own or if it is simply pushing the Earth leakage current over the point where the RCD trips. i.e When used on its own it doesn't leak enough current to trip the RCD but when other 'leaky' items are used at the same time then the sum total of the leakages is enough to go over the tripping current. Some things are notorious for leaking currents to their CPCs, dodgy PC power supplies being a big culprit.