There are several reasons it wouldn't work. Japanese algae eating shrimp, both Yamato shrimp (often called "Amano" shrimp) and Minami (rare outside Japan), are cold water shrimp. The temps for a betta are too high for them. They start to weaken and die off at the proper temperature for a betta.
Another is that they will be eaten. The largest Amano might be big enough not to be eaten, but there is no guarantee that a hungry betta won't tear bits off and kill the shrimp. The smaller ones are not any bigger than the CRS.
You would actually probably have more luck making a very dense area of plants and moss (dense enough that the betta cannot get into it) and going with CRS. They do well in that temperature range and will hide in there...
But still, at best they will only eat left over food and algae. Nothing will eat the feces, so you will always need to clean the water to get rid of it.
Which comes to another reason it wouldn't work. In an unfiltered tank, you will need to do frequent large water changes. Shrimp (all varieties) do not cope well with water changes. They are very sensitive to changing water parameters.
I have Minami Swamp Shrimp in one of my tanks (not betta) and I can only do maybe a 20% water change twice a week before the shock is too much for them and they start shedding their shells and dying. And this is in a well filtered stable tank.