Does he have any other symptoms, I couldn't tell from your video.
-The trying to get into small spaces, is that new?
-Have any of his living arrangements changed? (IE, different tank, different location, new light, loss of light, new plant, new ornament, ornament taken out, new filter...anything moved...ANY changes at all?)
-Is he eating normally?
-Are his fins clamped?
-Is he at his usual activity level?
-Does he "flick" his head or his body when he swims? (rather like a "hair toss" on a human, they flick from side to side, also called flashing.)
If all of his other behavior is the same as usual, and the only difference with him is the enlarged gills and heavy breathing, he probably has gill inflammation or scarring from high ammonia. I've got a guy who looks just like this. He eats, he swims, he's active and happy, but because of his poor pet-store care before I purchased him, he has permanent scarring on his gills, one side hangs lower than the other (the more scarred side) and he breathes heavily. He will always have it, and it's unsightly, but not life-threatening.
One thing that will make his life better, is getting a slightly bigger tank, and cycling it. (Most people will say that it's only possible to cycle with 5g and up.)
With an uncycled tank, even with regular proper water changes, there is still some ammonia almost all the time (the only time you'll be at 0 is *right* after you change the water 100%, and you can be as high as 0.25 or .5% ammonia before the water changes), and this constant ammonia (even in low amounts) will just exacerbate his gills. By getting him into a cycled tank, your ammonia will be at 0 all the time, and this will make his life better and his breathing a little easier. :)
You can try to cycle your 2.5 g, there are people who have done it, (I've never tried, so I can't comment from personal experience) but apparently it's very hard to do, and the cycle is *very* hard to get stable.
If you are on a budget or tight for space, you can usually find small, plain, empty 5g tanks at petco, petsmart, or walmart for under 12$. They're not hardly any bigger than your 2.5, but they can be cycled. Another option is to get a jar or Rubbermaid container that can hold 5 g's, and use that. :) There are lots of options, and they all work. :)
As for the filter, I've used the Aqua Tech power filters and never had a problem. They are 10$ for the 5-15 gallon size at walmart. Just get some carwash sponges (they're big, you can cut them with scissors, and you cannot use kitchen sponges because most have antibacterial/antimicrobial treatments on them that totally defeat the purpose of the filter), and sponge the intake and outflow (message me on here if you need information about sponging them) to make the flow really soft and to make sure your betta doesn't get sucked up into the filter.
Or, you can make a sponge filter. Here's a link to sponge filters:
Then you can cycle your tank. Do you know about cycling? If you don't, here are some links:
Again, if you need more information, PM me and I'll help you out. :) I'm not an expert on cycling *at all*, but I understand the basics, so I can at least get you started. :)
For a "lid" you can use binder clips and saran or plastic veggie wrap. Just stretch it across the top of the tank, and secure with the clips. Just leave a strip about an inch wide at the back so that air can get in and out. This is the cheapest method.
If you want to spend a little more money, you can get a versa top or canopy for about 7-10 dollars (most fish stores carry them, or you can order online), or a hood (which is quite a lot more money). Or, you can get a piece of acrylic or plexiglass at your local hardware store, cut it to fit (again, leave a bit on the back open for your filter or any cords/lines coming out the back and for air) and lay the acrylic across the top of the tank. This is cheap, usually about five or six dollars, sometimes less depending on the size you need and what your store charges.
I'm sorry for such a long post, but I hope that helps!! :)