Shrimps will do well with just a sponge filter operated with a air pump, that's the very least you should have in the tank if you intend to keep the shrimps, if you don't intend to keep the shrimps, just feed them to your betta.
Sorry to hear that putting your shrimps with the betta didn't go well. My first betta couldn't have shrimps either as he would go after them. My current betta is ok though, he gets very close and watches them but nothing else?
It would be best to have a filter if you can get one small enough, and if it is adjustable have it on low flow because it is a small tank, the shrimps are fine in flowing slow water,! Are you planning to keep them in this tank long-term or is this tank temporary until you decide where to go next with them? It is possible that some may die from injuries/stress/shock etc, but hopefully they will be ok.