Ok. (sorry for leaving, I had to go to bed)
Do you have any live plants?
Can I have a picture of the 15gal and the current tank?
I would suggest getting some gravel into the 15gal, and the putting some live plants into there if you can. I would suggest Dwarf Hygrophila, but get something hardy, at least.
Squeeze the current filter sponge into the water in the 15gal. This will help get it started. Then put the filter back in the 5gal. (also, I presume you have a filter and heater for the 15gal? Also add in some water lettuce if you can get your hands on any. Or any floating plant (except duckweed)
Leave the 15gal for a day.
When you are going to move him over, instead, get a bucket (that has never had any chemicals in) and put the recommended amount of aquarium salt and water in there (the water should be from the 5gal) and put him in there for 10-15 minutes. Then put him into the 15gal.
At this point the 15gal should have:
- A filter (could you tell me what type) that's current isn't to strong.
- A heater (set to 78f)
- Gravel for the plants (has to be quite fine but if you can't get your hands on it just use what you have)
- Live plants (something green and hardy)
- Floating plants (water lettuce preferably)
- A light (for the plants)
Then do a water test (if possible) of the 15gal.
If there is no ammonia, then move him over.
Try feeding him some live or frozen daphnia from a fish store (live is best) as this does not cause bloating at all. Also try some mashed up peas, as this helps congestion.
I hope this helps!
(Also, if you can, get some beneficial bacteria to help jump-start the tank. The plants will also help with this)