Anything over 250g requires you to fly with a registered adult operator, you can not fly any 250g + plane on your own, because you are too young to be an operator, you can be a pilot. Your BMFA A licence only exempts you from the dumb online test, it doesn’t mean you can fly alone. It also means you will have to have an operator register in order to get the operator ID you are supposed to put on your planes somewhere.
If you are flying in a sensible place and being responsible the chances of the police rocking up to check your licence are virtually nil however.
A club may provide the operator at the field for you but they won’t be able to do that for you to get an ID number.
There may be changes as we go along as well. As
@CarolineTyler suggests actively attending your local club will now be the best solution to your problem, as they will clarify things and their field will be ideal for flying gassers on.