In a vase that small, you don't need a filter. Just do water changes 1 time a week if its a gallon vase, or 2-3 times a week if its more like a 1/2 gallon. I don't suggest you keep a pet betta in anything less than a 1/2 gallon. If you get a larger place for your betta in the future, keep in mind that you will have to set the filter to the most gentle current, as bettas are not strong swimmers and need a place to get away from the water flow. If you get at least a 2 gallon tank, you can get a whisper mini filter for 2-5 gallons. It has a gentle flow feature that works well. Or get a gang valve that allows you to control the air flow to attach to some tubing and use a corner filter.
As for putting the vases together, being in constant view of each other might stress them out. It depends on the aggressiveness of each male. As long as there is a place for each of them to hide from the other's view, they should be fine.
Alexa