When i last had a need for filton on a saturday earlier this year it was closed and there was no special access priviliges via the flying club. They are easy enough to phone.
There is so little traffic at filton these days it only classifies as a big airport because the runway is long and wide.
There are four spaces on the south apron i always found you got one if there was one free and no priority traffic. If not the nortn apron is fine and they send transport. I cant recall ever having a marshall in 20 or so visits.
It is a fair way to the zoo. Bristol i would guess is closer but two or three times the cost. I think it was £100+ last time i was in there in a twin. Mind you you are handled very nicely.
The zoo (i have been once) was excellent, i enjoyed the visit.