There's a few possibilities for the problem that I can see
1) Chlorine burn
2) Ammonia burn - if you've not been dechlorinating your water, it may well be that your filter has not had a chance to grow the beneficial bacteria it needs.
3) The betta - some bettas can't tell the difference between themselves and guppies - indeed some bettas are just so feisty that they don't tolerate any tankmates.
My instinct is that the betta hs taken exception to the guppies and managed to catch this particular one a bite on the back, which has then been exacerbated by chlorine and/or ammonia.
The paper test strips, I personally don't like them at all, they are notoriously inaccurate. The Tetra ones don't have a test for ammonia on there, which, IMHO, is the most important thing to be able to test for.
Definitely buy a water conditioner aka dechlorinator product. The API one seems a good one, although I have never used it. If they have Prime (which sadly I very rarely see in shops), that's the one that I use, as do most of the experienced aquarists on this forum. Thoroughly recommend it, but if they don't have it, the API will be perfectly good.
Definitely buy liquid-drop tests for ammonia and nitrite. You can get them singly, I use the Salifert tests, usually found in the marine section, but they can be used for freshwater. If not, the API tests will again be perfectly acceptable. If you have enough money for a Master Test set (which includes pH, nitrate, and others, as well as the ammonia and nitrite), then so much the better.
When you get home, work out how much of the dechlorinator you need for 5 gallons of water, and put it straight into the tank. Then work out how much you need for 44 gallons, and put it straight into the larger tank.
Then use the ammonia and nitrite tests (on both tanks) and pop the results up here, and we can advise what best to do next.