Is it possible to upgrade a CO2 pellet rifle from 7ft/lb to (just under)12 ft/lb, the UK legal limit?

No, sorry. Co2 expands at a certain rate. The max power your going to get with a 177 is 700fps with 7.0gr for a single shot. The CX4 carbine has gaps in between the mag and barrel which leaks out co2. Buy a spring gun (RWS 34) and move out or help improve gun laws in the UK.