Your original idea with the 4 core cable should work. Personally I would use 5-core so the circuit back to the immersion in the house has its own neutral. You will need to take the immersion circuit out of the CU in the house and connect it to the 5 core with an MCB in a separate enclosure suitably labelled or a fused spur box.
You may need a separate earth rod for the Garden Room so it is TT earthed, rather than exporting the earth from the house presuming that is T-N-CS.
Sharing neutrals in this situation could very easily lead to tripping RCBOs or RCDs