If you don't have a filter in the betta tank, there won't be any beneficial bacteria or at least not enough to matter. Don't worry about moving gravel. The best thing to do is just start testing the water daily for ammonia and nitrite and make water changes (about 20%) as needed (up to 2 or 3 times a day if necessary) to try to keep the ammonia and nitrite under .25 ppm. With only 3 tetras (what type are they) the ammonia shouldn't jump too quickly. Feed them lightly also. The less you feed them, the less waste they will produce and the less left over food there will be to decay and turn into more ammonia.
Also, since the betta tank isn't filtered, you should be doing 100% water changes on that tank at lest every 2 or 3 days to keep the ammonia level safe there. And only feed him what he will eat. Don't put a lot in the tank and have some of it go uneaten and rot.
