Well, if you're doing frequent and large enough water changes on the small bowl, there won't ever be high levels of ammonia. If you move the fish to the new tank as soon as it's set up, you'll still have to do frequent large water changes to keep the ammonia levels down while the tank cycles. This will likely make your cycle take a lot longer.
I suppose it's really up to you. If it were me, I would just leave the bettas in their bowls and keep up with those frequent water changes while the new tanks cycle. This way you can do a "fishless" cycle on your bigger tank that is often shorter. Or, you can move your betta to the new tank right away and cycle the tank with the betta in it. Like I said, you'll have to monitor those ammonia and nitrite levels like a hawk and do big water changes to keep those values as low as possible in order to avoid harm to your fish. Either way, in the end you'll end up with a happy betta in a cycled, healthy tank.