I agree with Kelly, bettas are not suitable for that tank, a member on this forum was in a similar situation. There is not enugh gap in the surface for the betta to breathe.
Can you explain this "gap" thing further? Do you mean, "gap between the water surface and the lid?" (there is a few inches, and the lid has a screen that's open to the air). Or do you mean that the square opening up top is not large enough in area? I'm puzzled by this gap thing.
I can tell you that I have a betta in a fluval edge with a couple of little panda corys and my betta is doing great. I had him in a half gallon betta dish and felt bad for him within a couple of weeks of getting him, so I bought him the edge. He's loving it. He's active and healthy, swims around a lot and comes to the glass on all sides to follow me around and see what I'm doing, and he goes to the top for little gulps of air from time to time. It's just a 1 inch fish -- the opening is about 6" by 4", and there is plenty of air space above the water surface -- more than enough for my betta to access air.
The only negative is that he keeps depositing little bubbles all over the surface of my tank against the glass, so I have to keep using my algae magnet to clear the top if I want it to look nice . On the other hand, my betta loves following the algae magnet around so it seems to be a game for him. I assume that an active, curious fish is a healthy fish.
Anyway, the fish is doing great so I am curious why you feel it should be having problems breathing and what the relevance of "gap" is.
I should add a note to this:
For the first half hour or so that my betta was in the Edge, he'd go to the top, find there was glass instead of surface, and seem to panic a little. He'd swim about the top rapidly looking for the surface until he'd find the good-sized opening in the middle back.
I felt a little bad for him, but he got it all sorted out in no time. He's been in there three weeks and has not shown any sign of this problem since that first half hour. In fact, when I reach for the food tin, he goes right to the surface space to wait for his food. I'm satisfied that he knows his way around now.