Snowy,
As a former hill-climb competitor myself in the road car class, with my Cayman R 6 speed manual, it would be great to see a GT4 compete on the hills.
Jeff has already mentioned contacting Steve Kevlin, PCGB Motorsport Manager. He will advise you on the PCGB involvement in hill-climbing.
Alternatively, join a motor club near your area which organises hill-climbs according to Motorsport Association regulations (MSA). There are strict rules on eligible tyres for road car classes, along with mandatory safety equipment. Your car will need a timing strut as described in the MSA regs, fitted to the front bumper where the number plate routing is located. Andrew-CS has given you some useful links in this regard in a previous post.
On general note, most but not all, hill-climbs are held over a weekend, Saturday and Sunday, with at least 2 practice runs and 2 official timed runs in each class per day. It's usually an early start with signing-on and scrutineering taking place from 08.00am onwards. The day usually finishes around 16.30-17.00. Camping facilities are usually provided at most hill-climbs. Portable loos are also usually available. When I used to trailer my hill-climb racing Imp, my late wife and I used to stay in nearby B&B accommodation.
In my long experience of hill-climbing events, there is a fair amount of hanging around. Delays caused by "offs" when the track is red flagged, often impact upon the programme timetable. Delays usually reduce the number of practice runs allowed, but you are guaranteed at least 2 practice runs.
I would guess your GT4 will probably be shod with Michelin Cup 2 tyres. These will do perfectly well for hills both for dry and damp/wet conditions. In my experience the tracks were usually well drained, so pools of standing water was rarely an issue.
Good luck Snowy with your hill-climbing plans. Keep us posted on your progress.
I'll leave you with a parting shot of my Cayman R in action on the Fintray Hill Climb in 2014.
Brian