Ghosties are the most recommended because they are the easiest to find and tend to be really cheap in case your fish decides to make them into a snack! Cherry shrimp are another option, but you would likely have to order them online, and again, they frequently become snacks, albeit a more expensive one. I prefer to keep groups of 5 or more shrimp. They have a very small bioload, so you can easily have 20 or more in a heavily planted 10 gallon. Don't go over 10 though if you don't have a lot of plants. Start with a smaller number and work up to a higher one too.
They are also the most extreme escape artists I have ever had. While you need a top for a betta, you absolutely need one that covers every nook and cranny for shrimp. Obviously not air tight, but close enough to it.
Though really I think the issue here might be overfeeding. There should be no extra food sinking to the bottom after a feeding, especially not with pellets. 3-8 pellets (highly dependent on size here) twice a day is more than enough to keep a betta going. And unless you are using a pellet like the NLS Small fish formula, that number is really right around 3-4 pellets per feeding. Obviously you might like to have tank mates for your bettas because they can be very enjoyable to watch, but they should be looked at as additional fish in the bioload (meaning increased water changes) not as your clean-up crew. Even if the fish (frog, snail, shrimp, etc) eats something, they still poop it out later and overfeeding always = overfeeding and decreased water quality.
Though if there was a species that was the designated clean-up crew it would probably be shrimp...
Remember that the tank your fish lives in is his or her's whole world. Ask yourself this question, and answer it honestly: would you choose to live there?