Hi there, I'll help if I can...
The first quesitons I have to ask: Are the scales on the fish raised (sticking out/lifting up)? Is the swelling on just one side or all the way around the belly area of the fish?
This could be a number of things, but I'd like to warn against experimenting with meds if you really want to save your fish. Bettas are very sensitive to most medications. Also, remember, you can't medicate properly if you don't know what the problem is first. "Guessing" and just trying various meds is a good way to kill a fish.
This could be the beginning stages of dropsy, for which there is no cure, it could be an internal parasite, internal bacterial infection... and for each thing there is a specific treatment that is safe. A betta with dropsy can still be treated and can still live for well over a year.
OK, next questions... You mention that your water quality is fine, but didn't post test results. That, honestly, tells us nothing. When we ask for test results it's not just to know if they are within "safe" ranges, but also in how they balance out together. Knowing exact numbers is always important when asking for help, so can I ask you to please post exact results for ammonia, nitrite, nitrate, and pH?
As for the slime on top of the water, that, also could be a few things... the first thing that comes to mind is an overdose in water conditioner? When you do water changes, are you adding enough conditioner for the whole tank each time or just the water you are changing? Which conditioner are you using? Also, how often are you feeding and what foods besides the betta pellets? Are there live plants in the tank? Is the water cloudy? Are there bubbles or "foam" at the surface? What kind of filter are you using?
I apologize for the many questions, but I'd like to help, and without those answers, there's not much anyone can do. The more you can tell us about your tank and situation the faster we can help. Asking these questions all at once saves precious time... so I am sorry if this seems overwhelming.
And lastly, the food issue... most bettas will eat limited types of food. I can count on 1 hand the number of bettas I have seen eat the flake food over a 10 yr period of time. If you're looking for an alternative that the neons can eat too, try things like live or frozen adult brine shrimp and freeze dried plankton. The pellets contain everything a betta requires to meet nutritional needs, but a variety in foods (different food each day, not all different foods in one feeding) is always the healthiest way to go. The neons should eat all of the foods I suggested, which could save you time, money, and worry about who's eating and who's not.
I hope this helps, I will watch for your reply with answers and do what I can to help your fish.