1 gallon is certainly bigger than the cup the bettas are kept in in the store. The difficulty lies in maintaining good water temp/conditions in such a small space.
I kept my Sephiroth in a 1 gallon temporary unfiltered "tank" when I first got him (live in tropical area, no need for heater). He was tiny, and I didn't have another 2.5g to move him into. He stayed in the 1g for about 1 month. Even though I fed him sparingly, it was very hard to keep the tank clean - I would recommend you keep the 1g bare bottom with several live plants to help water quality. Java Fern is very easy to keep and inexpensive. Plus, your betta will really perk up with live plants in his tank. The aerator is not needed, as betta breath air via their labyrinth organ.
Bottom line, if you are diligent about water changes (50%, 3x/week minimum) and do not overfeed (2 pellets/day), and work to get the 10g within the next month or two, then your betta should be fine temporarily in the 1g.
edit: Look for the Azoo Palm Filter - it has adjustable flow, and I've used one on a 1g hex with no problem.