NO, NO, NO!!! This concept is insane! Yes, a Betta can live like this, for a few weeks, before it dies. A Betta's normal life span, if properly cared for, is 5 years or more!
Bettas have what is called a labyrinth organ. It allows them to breathe oxygen from the atmosphere when their bowls get toxic. That is how they are able to survive in little bowls for a while. Cut off his access to the air, and he is forced to stew in his own waste. It would be like you living in a tiny bathroom where you couldn't flush the toilet.
I don't know if Ivy is safe for them or not. I know some varieties are poisonous to cats, so I wouldn't try it.
If you want to be a kind, responsible, Betta owner, get up off the $20 to get him a proper home. He needs a minimum of a 2.5 gallon tank (bigger is better), with a heater and gentle filter. Put some real or silk aquatic plants (no plastic) in the tank for him to sleep and hide in.
I suggest you go to a book store or library and get a book or two on Bettas. You have a lot to learn about proper Betta care.
Good luck!