I can't help you much with the use of Coconut Oil, but I can advise on quality.
There are quite a number of Coconut Oils on the market and price ranging from very cheap to expensive. What I have found out is the cheap coconut oils have been rendered using bleaches and chemicals to remove the odours caused from mold in the processing of the oil.
This leaves what I call a "garbage" product that should not be used. This product does not taste or smell like the coconut oil is supposed to. Doesn't even make a good axle grease.
The more expensive ones are pricier but well worth the money. They are Virgin Coconut Oils and are cold pressed. There are several on the market.
The one we found that sold best, and which we use ourselves, is called
Alpha DME Virgin Coconut Oil. It tastes like coconut. There are other ones out there that relate in quality to this one.
Maybe the problem with the cat could be that if it's the cheaper oils, it doesn't have any smell or taste and could have a chemical residue that is bothering the throat.
Something else that you might want to try to help relieve the pain in the gums is to lightly massage the cheeks with just enough pressure to stimulate the blood flow in the gums, bringing in fresh blood and removing infection.
That web site that rustycat recommends looks like it could be quite interesting.
Just a note - I had a price quoted to me by a vet for one tooth removal and it was $750. so you don't really want to go there if there is an alternative.
Animals are such agreeable Friends.
They ask no Questions. They pass no Criticisms.