A visitor's Permit is easy enough to get. It costs £12 and takes a couple of weeks to sort out.
You need one to travel around with a gun but if you are travelling and using a gun with a SGC holder you don't need a permit as he can supervise you.
While travelling with guns, they must be locked in the car out of site, ideally with the forend removed and a wire trigger guard lock securing it to the car.
Ammunition must be kept out of sight.
Pigeon shooting is great fun. Arranging it is not a problem. Guides will charge between £60 and £200 per day depending on what equipment they have to supply and the place/time of year.
Best pigeon shooting is August, Sept, Oct and some good sport can be had in Feb and December if the weather is cold and there are a few shooters spread around.
If your time is limited it may be worth your while getting someone professional to take you out - no guarantees but likeihood ogf a lot of shooting definitely enhanced if you go with someone who knows what he is doing.
Imagine a Brit turning up in the US and going off to hunt a turkey on his own. Better to go with a local if you want to succeed.
If you are in the London area and I'm free, I'd be happy to take you out and you are welcome to use your choice of gun from Purdey hammer gun to Winchester pump.