Hello, I'm Dr. Deb.
I'm so sorry that Jack is having these issues.
The short answer to your question is that, yes, it's possible for trisodium phosphate (TSP) to cause oral issues if enough was ingested by a cat. It wouldn't necessarily be toxic for them but the damage that might be done could be fairly significant.
This particular product is very alkaline which can be quite corrosive although sometimes it can take several hours for the burns to appear in the mouth. If oral lesions are present, then there might also be esophageal or stomach lesions; if no oral lesions are noted, then the stomach and esophagus are not likely to be affected.
Initial treatment consists of water or milk but if oral lesions are present, then additional treatment is usually required: carafate slurries, pain medication such as Buprenex, antibiotics and acid reducers. If the oral lesions are severe enough, some cats will need to be force fed or have a feeding tube placed.
One over the counter option for Jack would be Pepcid AC (Famotidine) at a dose of 1/4th of a 10 mg tablet. To avoid further injury to his mouth, I'd crush the pill, mix with a small amount of water or tuna/clam juice and use a syringe or small turkey baster to dribble this mixture into the side of his mouth twice a day. Even if your vet gave other anti-nausea medication, it would be safe to also give Pepcid AC at the same time.
Chlorine might also cause oral lesions although typically it's diluted so much with water that it's not a problem for most cats.
I hope this helps. Deb