Have you tried using denatured alcohol to strip the finish? If the finish comes off easily with alcohol, it very likely has a shellac top coat at least. Nitrocellulose lacquer is the other common top coat finish. Alcohol will sometimes dissolve it slowly but lacquer thinner will remove it and shellac very easily. You can see a list of ingredients in lacquer
here. Castor oil stands out as an ingredient that could cause an issue I suppose but I do not know if it all evaporates or stays as a component in the final finish. The resins used in lacquer might be worth some additional research related to toxicity and reactions.
Have you ever had a reaction to pine? I have heard of people suddenly developing an allergy to pine. I think that a lot of varnishes were made with a pine resin, also known as rosin or pitch. They may have been dissolved or carried in turpentine which is also derived from pine.
Edit to add: Are you allergic to bee stings? Bee's wax was another common finish at one time but I would doubt that was the primary finish on your piece.
Unlikely on your piece but Urushi from Japan is a finish derived from a plant related to poison ivy. I can imagine that sanding it could require caution?
--Nathan, TX. Hire the lazy man. He may not do as much work but that's because he will find a better way.