The water going in definitely needs to be treated. What size is your tank? How big is your smaller tank? Let me first say I'm not saying what you are doing isn't possible, but it's a challenge to keep the water chemistry in line. You are trying to keep enough ammonia in the tank to get the cycle going but and at a "safe" level for the fish. Not to be harsh or arrogant sounding but if you're overdue on the change it could already be too much for the fish to stay at in that tank with a level of .25 ppm of ammonia. Even if you kept it right at .25 ppm which is what most people recommend as the limit for "fish in" cycling with Bettas it will cause your Betta stress over time. Then add a few spikes on top of that and you have a vulnerable fish to say the least.
I understand your dilemma with the tanks and such. I would highly recommend to be looking at a Pet Keeper and a 25w heater as a back up just in case the fish starts to show signs of distress. At the earliest sign of distress such as lethargy or fin splits I would move the fish out and switch to fishless cycling. This will give the best chance at recovery and a shorter healing time.