PRS are assignees by virtue of membership of composers or publishers of the performing right in words and music of songs, and music of tunes. Such right includes the right to perform in public. Don't listen to people who try to define you so that you aren't "public". The argument does not win in court.

Most composers and publishers are members. Some composers reserve the right themselves to perform their own songs. Some words and music are not protected by UK copyright. Some composers are not members of the PRS (or overseas affiliated societies) and if they are published there is a very very small but finite chance that their publishers are not either.

Subject to this, the PRS are right to collect the royalties because they are the collecting agents of the authors and rightsowners, but most pubs have PRS licenses any way to cover their jukeboxes so no further license would be needed (this may depend on the exact tariff).

I suggest you talk (rationally) to the PRS direct. I have always found them very helpful. You can get copies of all of their tariffs from them and the person who dealt with the pub may have applied the wrong one. I am not sure how much discretion they have to waive or reduce tariff fees outside of litigation for back payments.

If the pub needs a PRS licence anyway (and most do) why won't the landlord cover the cost or part of it out of the profit he makes on beer sales to you?

Sorry, no magic exemptions.

Usual disclaimer about no solicitor-client relationship.