Using the internal 1.1V voltage reference would improve things. The accuracy would then be on the order of 1mV per step, or 0.1 degrees, though I would still not consider this a precision voltage reference (nominally 1.1V, but can be between 1.0 and 1.2V per spec). However, the temperature control loop is still a problem. The heater consists of nichrome wire embedded in fire cement. This has a certain amount of thermal mass and will continue heating (i.e., raising the temperature) of the water bath and developing tank after the heater is shut off. Just how much is unknown without actually building it and trying it (or doing a whole lot of math that I don't care to do), but the control loop simply waits until the actual temperature exceeds the desired temperature before shutting off the heater.
I've read of people using Auber PIDs with temperature probes that can read in 0.1C (so they say), all connected to heating elements via a relay. The PID apparently "learns" by repeated cycling how much voltage to apply and for how long, in order to hit the desired temperature. I've priced these, and the PID and probe are around $60, while the heater is under $10, the relay being similarly priced.

Perhaps that would be a good alternative?