Can I be a member of more than one local club?

Yes, but in BridgeClubLive you can only register with one Club at the same time.

The Local Club system is to help you find other members of your local club and play bridge with them
If you arrange to play with Club A on a particular day/time.
Then make sure that you are registered as a member of Club A on that day/time.
Once logged into the playing area of BridgeClubLive then press the [Local Club] button middle top.
You will then see the list of Club A members, which room they are in, etc. and also if they are offline.

The next day/time you may have arranged to play with Club B.
Again make sure that you are registered as a member of Club B on that day/time.

If you change which club that you are a member of, and you are already logged into the playing area of BridgeClubLive, then use settings/other to refresh the Local Club List data