If I record the songs playing & they sell bootleg CDs, I don't have a defence of "but it was played in public, so all copyright was waived". That's not how copyright works.
If the musician gives someone consent to record (plus subsequently use the material in various ways) then in fact they do waive some of the copyright with regard to any of their original songs. The consented party has been given an implicit license to the material within the bounds expressed in the consent.
The status of the licensed songs is not affected whatsoever, and the musician has made no legal representation in regard to that. The musician is paying the royalty fees to PRS and doing everything they are supposed to.
The musician can say, "you have my consent to record and use the recordings, but be aware that some of the songs are copyrighted by a third party and I'm using them under license. Nothing I say gives you right over those songs."