I seem to remember Betta Bulbs are a generic brand name for various species of Aponogeton. These plants grow quite well without special care and low light.
I think they might grow too big for a 1 gallon tank though. The leaves of most of these plants are 15cm+ each and the plant will have several leaves. I had an Aponogeton ulvaceus in my 7 Gal betta tank and had to keep trimming the leaves because it got too big. Also some of these plants might start to look not so good after 6-9 months and need to go through a dormant period to simulate winter. You have to take them out of the tank and keep them in a house plant that stays quite moist. After a couple of months you can put it back into the tank and it starts again.
It depends on the bulb and the light allowed. If your bulb is in lower light, it will grow more slowly, and the leaves are smaller. Again, depends on the bulb.