If you add shrimp, I don't think you need a bottom feeder. They do a pretty good job of cleaning the bottom in my experience once there are enough of them.
On the betta, neon tetra question: make sure there isn't too much chasing or nipping going on. Also, watch the betta for fin rot or tears in his fins. That could indicate a problem. Also, if the betta seems to spend a lot of his time hiding, that would indicate stress.
I would plan on rehoming the tetras to your new tank whenever you get it. The betta will love having the tank to himself. My betta shared a 10G with six Harlequin Rasbora for a while and he seemed ok but was hiding a lot. I moved him to his own 5G and he's been much happier and bolder.