This is what I was afraid of
You don't have a fat axolotl. You axolotl is retaining fluid.
Fluid Retention is "Normally" caused by kidney failure. I have one melanoid who is only 2" long suffering from it, I already have euthanized two other of my babies because of it. Though this guy has been like this for almost two months now, I just cant euthanize them until they stop eating, which this guy is still eating.
My babies who developed it it was probably caused by genetics and not any kind of sickness. but since your guy is older/bigger I suspect he is ill somehow.
Caudata Culture Articles - Salt Solutions in Treating Salamanders Here is the article which talks about fluid retention. There really is not much info out about it. It states to ONLY DO IT UNDER DISCRETION OF A VETERINARY AND IN CONJUNCTION OF ANTIBITIOTICS! I tried already without antibiotics and without a vet and he did not make it through the treatment! but it just gives you an idea.
Like I said I don't know how these guys fare, I dont find much information about it but like I said the smaller ones(10-15days old) stopped eating after about 10days so I euthanized, the guy I have right now is 4months old and has showed signs of this for 2months now and is still eating and pooping, though his belly is so big it limits mobility, it really is sad to watch
AHHHH EDIT: I just saw what you have as substrate! You should really remove those stones! axolotls can swallow them and they can become impacted!
Edit edit: I mean there could be other reasons for whatever is going on but like I said I have had no luck in getting much information on fluid retention in axolotls and really any caudates at that. I'm sorry I cant be more of help but just telling you your axolotl is no getting fat. Still feed it and take care of it like you normally would. I would suggest bringing it to a vet if you can but I know its not always plausible. but as far as I have tried, I tried the solution in the article, I tried a normal 10-15minute salt bath, I tried fridging, and it just gets bigger and bigger.
I am pretty sure a vet will puncture it to remove the fluid, start antibiotics to keep infection away and then the salt solution is to help it from coming back. but thats just a guess and not a fact.