Its extremely common after reading lots of stuff here from the many experienced fishkeepers to realize with disappointment the your filter designer was focused much more on how to get you back to the retail isle buying replacement cartridges than he/she was on designing a good filter for you and the fishtank! Most hobbyists don't know how to go about optimizing their filter purchase or their filter media choices until about the third time around probably!
But people have come up with lots of creative fixes and improvements once they've realized all this, so there may be ways you can get a lot more detailed about knowing what media you've got and how it could be replaced/optimized within the filter box you've got. At the easy end of things, sometimes there are multiple types of cartridges that fit and one or two of the choices are of a better biomedia type. Or, with more difficulty, I've read of people slitting open the containing part of various types of cartridges and replacing the contents with media that's different from the original.
The really small filters for betta tanks are kind of a special case. There are bunches of betta lovers here on the forum and lots of them have experience with the few really good little filters out there. Hopefully one will come along for you!
Carbon is not primarily a biomedia. It crumbles over time and goes out with the water changes, taking its bacteria with it, unlike the preferred biomedia choices of sponge and ceramics. Carbon's primary use is to be kept on the closet shelf in waiting. If you have a situation where medications need to be removed or wood tannins cleared or some strange organic smell must be gotten rid of, then carbon is your friend, but it only last 3 days max and then should be removed and trashed. In this correct role, its known as a "chemical filtration media."
~~waterdrop~~