Glad to hear you are getting a bigger tank. Since I really enjoy keeping Bettas, I have to say this. No fish was meant to live in a 1 gallon bowl. Bettas need at least 2 gallons to live a happy and healthy life. You can even go up to 10 gallons for 1 Betta. I have yet to hear a Betta fish argue with a bigger place to live. There, now I'm done giving my speech on bowls and Bettas.
Bettas are fish that like calm and quiet water to live in. The aerator stone may have been causing to much disturbance for him in such a small space. Since Bettas will come to the top of the water and take a gulp of air to occasionally breath out of water (this is normal as long as they aren't gasping), make sure that the widest part of the bowl is where the water is filled to so that he has the most surface area to get air. He shouldn't need the "bubble stone." Make sure that you watch ammonia and nitrIte levels so that they don't get too high. If they are higher than 0, that really is too high. Do a 25%-50% water change to decrease the levels and also the stress on your fish. Check those levels at least every other day with a 1 gallon tank. JAVA MOSS will also keep ammonia and nitrIte levels down and you can purchase it at an online store called Aquarium Garden. It also provides a great blanket for your fish to rest on.
On reflections of himself, this is stressful for him if it is lasting longer than a minute or two. I would get a rectangle tank for him ASAP so he doesn't get over stressed. When you put him in the new tank for the first time, add 1/2 TEASPOON per GALLON of AQUARIUM SALT. This will help increase his ability to utilize oxygen as well as protect him by giving him a thicker slimecoat. Don't "over salt" your fish, so if you choose to add aquarium salt on a regular basis, do it on every third water change since salt doesn't evaporate or get filtered out.
Hope this helps! Let me know if you have any other questions!! There is a link below on new tanks that you may find interesting, too.