[Diana]

Sponge filter has to run 24/7. That is where a lot of the bacteria live. They need a very good oxygen supply.

Have you looked into the smallest Aquaclear? There is an Aquaclear 20, that I have run on a 10 gallon. It is adjustable. 100 gph max, and it can be adjusted to about half that. You can slip a cube of sponge over the intake for added filtration. (Cut a slit or X through the sponge, leaving just the bottom intact and slide it on. You can use an Aquaclear sponge or a slightly coarser one.

If you have that much mulm building up there is something wrong with the set up. The filter is supposed to pick up the mulm, so it is good that it was getting into the zoomed, but bad that there was so much it was clogging the tubing. That is a lot of mulm.

If a filter runs too slowly it won't pick up the mulm. It won't clog, and the Betta will be fine with the slow water movement, but you will be vacuuming the tank a lot.
If a filter runs faster it will pick up the mulm, and you could try placing it where the outflow hits a rock or driftwood and is broken up rather than making a strong flow that bothers the Betta.

 

[Diana]

I use a coarse blue sponge sold for pond filters or sumps.
It has very coarse cells so the water flows through it easily enough. It gets darker with use, or you could look for a black one.
I have not put anything in the path of the HOB filters. When the water level is correct they sheet across the surface of the water and are diffused when they hit the front of the tank.
I have kept several Bettas like that, and they do fine.