The obvious, and nearest, society is perhaps the Bristol and Gloucestershire Archaeological Society, Hon. Gen. sec. David Smith, MA FSA, 22, Beaumont Road, Gloucester GL2 0EJ. There is a web site, but I am embarrassed to say I don't know it's address.
Have you tried asking at the Museum? 
  ----- Original Message ----- 
  From: Kate Giles 
  To: Brit Arch 
  Sent: Thursday, February 03, 2000 9:12 PM
  Subject: Local Arch Sites in Bristol

  Dear all,

  Please could you supply me with any details of arch socs in the Bristol area, as internet searches have come up with nothing local except Somerset.

  Thanks in advance,