Help needed to verify a potential issue.

[Drum]

I have been running 2 ESP-12F modules on battery and solar cells, to test for the for the deep-sleep issue. When they were sleeping for 10 minutes there was no problem, but when I changed it to 1 minute the voltage started running to the point where it the modules stop functioning. The problem appears when the battery is charged again without disconnecting the power completely. The module is in some kind of a blocked state, which does not reset without interrupting the power.

Does anyone have a good bench power supply, can verify this and test how low the voltage can go before the module locks up?

Also, I am looking at using a LDO with a shutdown pin to cut power before the module locks up and also to keep the battery from getting drawn down too low. Does anyone have an idea on a circuit on this, or a better LDO ? I am using MCP1826/1825 with shdn pins. I have not been using the shdn pin yet, it needs to be pulled low.

Thanks!