Is it possible for your to post a picture? I would think the brown can be his normal coloration especially if he always had it. The clear picture will definitely help.
What you don't want to see is when you Shine the light on the betta’s body: if it looks like it is covered with a fine gold or rust mist, then it has velvet. But i don't think/hope he has it because if he would have that untreated for a few days he would show more symptoms like darting,scratching the body on objects ., more coloration on the body, clamped fins.
Is it possible for you guys just raise the temp in the house up to at least 75*? See how he doing.
Yes buy spring water and do regular changes.But do 2-50% and 1-100% water changes a week. Or you even can do one full water change every about 4 days.
When you switch him from distill water to spring water do it slowly. Like i wrote before do 2-3 or even 4 water changes with spring water (about 30-50%). Just to acclimate him slowly to the new water.Not sure what is the difference in pH or other stuff that in the water so that is why i recommending do it slowly , so you don't shock him with the difference
Also when you do full water changes you can just let water sit in the room temperature for about 24 hrs and it will be the same temp as his. It just convenient for me at least and i don't have to worry to shock him with the drastic temp difference.
I think any pet store can check your water. Of course when you bring the sample do not put water conditioner in it.
I am not familiar with the water conditioner that you are using but if it removes Clorine,Chloramine,ammoia, nitrate,nitrite it fine . Just decided to google it and it's fine to use as dechlorinator when you use tap water.
If the store will tell you that your tap water has ammonia in it(some people has the problem) buy water conditioner Prime by Seachem, it the best water conditioner and helps in the situation if your water has ammonia in it. Unless you will decide to use spring water.
Food: If you will go to the store take the ice with you so you can buy him frozen blood worms. Usually you can find it in the refrigerator in the store. If you will not buy frozen ones you can buy
Best recommended pellets are:
Freeze dried bw you can use only a few times a week as a snack.
2.5 gall tank you don't have to cycle if you don't want to. If you don't cycle it you will need to do 2-50% and 1-100% water changes a week. Or again you can just do one 100% every 5-6 days. That is what i do. And if you don't cycle don't buy the filter. Filters are not necessity for bettas.
If betta will refuse to eat or will change his behavior even more(more lethargic, laying on the bottom, hanging on the top more, darting, scratching, clamped fins, more visual symptoms on the body) then there is definitely a problem that i hope with more help we can figure out.
The food that i gave you the links of you can find at the pet store but the stores i go they always out of it, and i have to wait another shipment. But if you have to drive far away may be they can call you when they get it or you can order on line. I have all of them. I just alternate between them. NLS ones sinking to the bottom very fast, the Omega one is not.