Hmmm, have you tried using a wire pan scourer?
They do great at removing gunk an they don't scratch they glass.
How many gals is the vase? Looks pretty cool for a small tank idea.
Bettas really need a filter and a heater to be happy and healthy. I would only use that vase if you can rig it so you can get both in there. They also need like 2.5g minimum. Some people say 5g minimum.
I think that vase would be amazing with plants and some shrimp