I am the monitor for this week and would be happy to help you with this inquiry.
Carpenter’s glue is a PVAC adhesive (polyvinyl acetate), much like the familiar white Elmer’s glue. Repairs made with this type of glue can be very difficult to reverse — and I can speak to this from personal experience! These glues do not have great aging properties, and actually become stronger over time. This is not ideal, as the glue may become stronger than the wood itself. As wood is a hygroscopic material, it will react to the surrounding environment by expanding and contracting slightly. If this happens, this super-strong join will not budge. This could introduce new stresses to the surrounding area and result in the formation of cracks. Plus, we don’t know what additives are used to make carpenter’s glue “acid free,” so it is difficult to know what impact these additives will have on how the glue ages.
Fish glue is also a hygroscopic material, and is reversible in water. Given the option between the two, I would recommend using fish glue — any future conservator who needs to reverse the repair will certainly thank you! Without knowing any details about the piece other than your brief description, I think it should be strong enough to get the job done.
I hope this is helpful. Please let me know if you have any additional questions or would like to share a photograph to review!