So after some digging and a few questions online, I've figured out that the power available is minuscule. Possibly on the order of uA(microAmps). Even running a single ATTiny85 connected to this power line without much more then a medium sized capacitor is not enough to keep it from browning out. Inspection of the existing board and processor indicates that the battery onboard, that I suspected might be for memory or RTC, is more likely for the system to support running the MCU and display for short periods of time. With deep sleep on the processor and a few interrupts to wake it up when needed, I suspect that the battery can charge off the trickle from the power supply and take up slack when the load it high.
This is, not suprisingly, well outside my knowledge area, but I think there might be some solutions that can be found by using the existing hardware as a reference. If we can put an ESP8266 into deep sleep(or another board if we have to, like a BLE based one), we should be able to wake up on input or packets to allow us to process data. We may have to use a bigger battery as a power sink though.
For now I need to do more research/testing.