Per the engineers at Wayne:
In theory this should work. In fact, one of the features that we built into the new ESP-25 is the elimination of the constant annoying alarm for lack of AC while installing the system. That alarm now only sounds if the power has established an AC connection and then it is removed either physically or by power outage. So if it is never hooked up to AC this alarm will be forever disabled. The rest of the board will not see the difference from a regular battery. The solar cells should work just fine assuming they are regulated and feeding a battery bank. The normal alarms when the pump kicks on and if the available voltage drops below the required threshold cannot be silenced. Just to be safe, if this type of application is going to be more than one or two units, I would try it out before committing to large quantities. There could be something in the circuitry I am overlooking that needs to see that AC signal, but my initial read is that the AC is only used for charging the battery. If you have any additional questions, don�t hesitate to ask.
p.s. This will only work on the new digital version of the ESP-25, not an old version or the ESP-15.