Without an ammonia source, the cycle will NOT continue. If you remove all the fish, you will need to add pure ammonia to continue the cycle (this is called fishless cycling, you can search here or on google and get tons of info about it).
You have a few options here. If it were me, I'd most likely just leave the betta and the frogs in the 10 gallon and do 50% daily water changes (or twice daily if needed) until the cycle completes and the bettas fins heal up. You said you have live plants, they will also consume ammonia and nitrites. So if you're inclined to, you might consider adding some more live plants, at least temporarily, to help things along. Once the cycle is done, you could move some of the plants to another tank maybe. I know any excuse to buy more plants works for me, so thought I'd throw that in there
I understand that water changes might be easier in a smaller bowl, but really 50% water changes in a 10 gallon tank aren't that much of a hassle. I'm not sure about using the bettafix in the 10 gallon with the frogs, but I'd assume the bottle should have some type of info on that? Or you might try their website. Usually, with frequent water changes, medications aren't necessary for a bettas fins to heal up.
Also, not doing water changes does NOT help the cycle progress any faster. Without water changes during a cycle with fish, the fish just suffer (or die) more. Not doing water changes during a fishless cycle can result in a pH crash which can stall your cycle or even kill off some of your bacteria. This is especially common when doing a fishless cycle with soft water.
You have a few options here. If it were me, I'd most likely just leave the betta and the frogs in the 10 gallon and do 50% daily water changes (or twice daily if needed) until the cycle completes and the bettas fins heal up. You said you have live plants, they will also consume ammonia and nitrites. So if you're inclined to, you might consider adding some more live plants, at least temporarily, to help things along. Once the cycle is done, you could move some of the plants to another tank maybe. I know any excuse to buy more plants works for me, so thought I'd throw that in there

I understand that water changes might be easier in a smaller bowl, but really 50% water changes in a 10 gallon tank aren't that much of a hassle. I'm not sure about using the bettafix in the 10 gallon with the frogs, but I'd assume the bottle should have some type of info on that? Or you might try their website. Usually, with frequent water changes, medications aren't necessary for a bettas fins to heal up.
Also, not doing water changes does NOT help the cycle progress any faster. Without water changes during a cycle with fish, the fish just suffer (or die) more. Not doing water changes during a fishless cycle can result in a pH crash which can stall your cycle or even kill off some of your bacteria. This is especially common when doing a fishless cycle with soft water.