The problem with agression to the female isnt exactly the fry. Before breeding, the male will blow a bubble nest ( betas are a member of whats called labyrinth fish, they have an organ called the labyrinth sack/organ that allows them to breath air, this is their primary mode of respiration, the gills do little work.) The male takes a few breaths of air, then blows a bubble out of his saliva and keeps doing so until hes created a bubble nest.
Then the male and female will breed, as the female drops her eggs and the male fertilizes, the eggs drop the to substrate where the male catches them in his mouth and places them in the nest. The male then chases away the female and guards the nest. In the wild the female would be long gone, but in a tank with little room to escape she would be destroyed by the male.
The only reason you dont technicaly *need* a filter is because of the betas method of respiration, however frequent water changes are suggested. If you'd add any other type of aquatic organism i would suggest some type of filtration. Considering the tank is so small either get a larger tank (10 gallon?) with filtration or purchase a small corner filter or mini undergravel filter system. Im not even sure if they make undergravel filtersystems for a tank that small.
As far as ghost shrimp go, I doubt the beta could even catch one