Under Copyright law, fees are payable to the owner of the copyright. PRS collects these fees on behalf of its members or their estates, and distributes them according to its rules.

Unfortunately, it is almost impossible to establish whether any fees are due in many cases, and their approach is to claim unless you can prove it isn't due.

Fees are not necessarily due if

a) the item is in the public domain OR
b) the owner of the copyright is not a member of the PRS or its foreign equivalents (PRS administers the fees for corresponding societies abroad)

The idea of swamping them with lists of (public domain) tunes played at each club and session might work, but would have to be well-supported, and might just backfire by raising the visibility.