If you want my honest opinion, I have to say that your 2.5 gallon tank is too small and unsuitable for any fish.
The minimal recommended fish tank size is 5 gallon in order to have stable water conditions, as well as to have the room to install proper equipments. So you are on a tight budget, but a 5 gallon tank only cost about $11 in most shops I have seen. A 10 gallon goes for only $14 here.
The basic set up for a proper betta aquarium should be:
1. A 5 to 10 gallon fish tank,
2. An adjustable heater
3. An aquarium filter
4. A thermometer
5. Water conditioner
6. Fish food.
Everything else is optional.
I have noticed you did not mention a filter in your post. It is the next most important thing aside the fish tank itself.
No filter = no aquarium nitrogen cycle, and the ammonia produced by the fish will not go anywhere if there is no nitrogen cycle going on in your fish tank. It will kill the fish for sure just the matter of time.
You also do not need the air pump. Betta has the labyrinth organ, which allow them to breath directly from the air.
Please reconsider your plan if you truly care about the well being of your next betta.
Once you have all the proper setup, then you can do a fishless cycle. Without a proper setup (no filter, too small of a tank), it will be quite impossible to cycle the tank for the fish.
