The tank size needs clarification as you stated a 5 gallon bowl (which I haven't heard of one this big) followed by small bowl. Did you mean a 1/2 gallon bowl?
Okay, first things first - he needs a heater. His water is too cold even for a healthy betta to be at optimum health, and certainly not one with compromised health. And that is compounded by the fact that one of the key elements in successfully treating velvet (if he indeed does have velvet) is higher than normal temps, in the mid 80's. The normal temperature range for a healthy betta is 78-82. The most important factor about proper temperature is it being stable with no fluctuations. Velvet is treated with with Cupramine (or Coppersafe I believe), high temperatures, and blackout (keeping the tank covered and blocked from light). If you have a 1/2 gallon bowl I honestly don't know if it is even possible to safely treat for velvet in such a small volume of water. The meds for velvet need to be used with extreme caution as a slight overdose will kill the fish.