I would certainly not try this. Leaving aside issues with the betta even if female, the badis is a shy and retiring fish at the best of times, and large tankmates is not advisable with such fish. Here is some data on the Scarlet Badis.
Origin and Habitat: Brahmaputra River system in Assam and northwest Bengal states, India. Found in shallow streams thick with aquatic or marginal vegetation and a sand or gravel substrate.
Compatibility/Temperament: A naturally shy and retiring fish, it may be maintained in a community tank of very peaceful fishes that are not active such as smaller rasbora (Boraras and Trigonostigma sp.), pygmy gourami, small characins, and any of the dwarf Corydoras species. Males may be aggressive particularly in small tanks.
Given its very small size--this is the smallest known fish in the order Perciformes--this species is very suitable to nano style tanks and small planted aquaria. In a 5 or 10 gallon aquarium either a pair or one male with 2-3 females is recommended, and in larger well planted aquaria a group of males/females. The males will form small territories, and plants and pieces of bogwood should be placed so as to provide suitable territory boundaries. The substrate may be fine gravel or sand, and floating plants should be present to shade the light. This fish remains close to the substrate.