You said you were not allowed to get a heater. Why is this? I really reccomend getting a heater. It will lower his risk of infection/disease. I have the hydor theo origional 25 watt. It has wonderful reviews, and so far it has been doing great in my ten gallon.
Also, I really think you should get a better tank. It sounds like you are doing the frequent water changes needed in a tank that size, but having a bigger tank will make sure that the water parameters are much less likely to fluctuate in parameters like ammonia or nitrate and temperature. It will also give your fish more space, thus reducing stress (which is a cause of tail-biting).
Many people believe that a 2.5 gallon or even a 5.5 gallon, and sometimes even a ten gallon, should be the minimum tank size for a betta. A 5 gallon has a very small footprint if you are concerned about size, and it would be great for your fish.