Switzerland are signed up to the ECAC agreement for permit aircraft, though this strictly only applies to short term visits.
The
LAA tech leaflet gives contact details for the FOCA officer dealing with those matters.
If the relocation is permanent, then they will likely require you to register it HB and get flight approval locally.
If you can stay on the G- then it's a good idea not to let the PtF lapse as the LAA can't authorise permit test flights outside the UK for a lapsed permit aircraft.