Hello, and welcome.
I'm sorry Henry has you worried.
You're right, that if he is hyperthyroid, these are some of the symptoms he would be exhibiting. Ask the vet if a thyroid panel was done, along with the other blood work, and if not, request it to be done at this time. Here is more information about hyperthyroidism in cats: CLICK
Because he's a longhaired cat, he might be ingesting a lot of fur during self-grooming, and sometimes, some fur can 'tickle' the back of the throat and cause gagging or coughing. In addition, he may have some acid reflux, which means excess stomach acid is coming up into his throat, causing a burning sensation/irritation and nausea, making him gag, but he has nothing to bring up, so it's just frothy saliva. Ask your vet if you can give him some regular Pepcid AC (Famotidine) (10 mg), just 1/4 of a tablet, 20 minutes before his breakfast, and see if this helps. Here is more information on this over the counter acid controller in cats: Pepcid AC
You can help him ingest less fur during self grooming by combing/brushing him a few times a day.
I hope all will be well with Henry!
Please be so kind as to rate my answer with positive feedback, as that is the only way I receive recognition from the site for my assistance. Thank you very much!
Providing a positive rating will not end our conversation, should you need me for any follow-up. Simply click 'Reply' for clarification or additional information, if needed.