armoured cpc problem

[andekoch]

I would bond the pool house to the earth spike and similarily connect the gas main and the armour cable to the same spike (via a met or the earth bar in the consumer unit).

I agree that there will be multiple parallel paths due to the armour and the copper gas pipe. The break in the armour in the joint works in your favour here.
Ideally the gas pipe would have a plastic joint inserted to isolate it from the main house.

Then treat the pool house as a TT installation.