Sky Diving club would be fun here, but as you said the red tape will be a nightmare.
As far as locations go, the South east is no good as you would get in the way of the Shaik Isa Airbase traffic.
The Center near Awali is no good as you are too near the Palace's, Riffa Heli base and the army base near the F1 track (with it's own small airfield, the Ex-Awali Runway also where that Vickers Vimy landed before they built the base.
The South West is no good as it is a nature reserve and host the Military training area, plus a few palace's.
The a lot of the south is still regarded as military and they are not keen on people going in there, even on foot.
Can't do it from the main airport due to Traffic and security.
Perhaps in the desert near Budaiya, but that desert is rapidly disapearing.
But what might work is perhaps on Hawar island, there is a long road suitable as a runway for small planes up to Caravan size I hear. And on the side you could utilise the aircraft to ferry people over there from the airport.
But your biggest problem will be the red tape and making sure your nowhere near any important facility or house. Perhaps the patronage of someone high up in the goverment would help things greatly, or perhaps a joint club with the military, I'm sure some of them are keen. I think there are even elements in the forces that are trained for sky diving/para dropping. I recall seeing some on tv once.
Remember as far as I know remote controled aircraft are still prohibited here.