If the copyright is managed by one of the agencies then it should be possible to obtain the appropriate licence from them. Isn't the difficulty here that the copyright isn't registered with an agency and the copyright holder is hard to identify?

Whether it's worth the risk of recording without permission does of course depend on your view of the possible consequences. If a claim was subsequently made then I would agree that the copyright holder is entitled to royalties at the appropriate rate, but where the artist had made reasonable efforts to find the copyright holder who had not taken steps themselves to make their ownership discoverable then I would resist any additional punitive claim. It's a matter of judgement, but the copyright holder also has to decide whether it's worth pursuing a claim as far as the court over a small sum (I'm assuming this will be a fairly small-scale release) or to negotiate a reasonable settlement. Again, I'm not talking about the agencies here but unregistered copyright owners. In my experience, amateur composers on the folk scene are often reasonable about these things. But only the OP can make that judgement.

However problems can arise if the copyright is subsequently assigned to an agency, even if you had the copyright holder's permission at the time of recording. My band recorded a tune by one of the band members, who held the copyright but is not a member of MCPS (the UK agency), and obviously we had his permission. A few years later it was recorded again by another band member for a solo album with an established label, and that label (which was registered) handled the copyright on the composer's behalf. MCPS then came after us for a fee, despite the fact that we had the correct permission at the time. It took some time to unravel, and then only with the cooperation of the label. I don't see any way to avoid this, although the lesson was to get all permissions in writing even where they are members of the band.