Indeed, using unprotected batteries on picocalc is a real hazard, especially if the 2 slots are used (there is no balancing circuit).
I only had unprotected cells in reserve, so I had to check to avoid any nasty surprises.
From what I’ve seen, the only way to charge batteries “properly” via the AXP2101 is via the Type-C port… But if you plug in the pi pico board, the voltage goes backwards switch U102, powering the AXP2101.
And if, in this configuration, you “switch on” the picocalc using the usual button, hop USB voltage directly to the batteries and happy fireworks 
Unfortunately, the picocalc lacks circuitry to switch/protect the VSYS coming from the pico…
So if you’re doing dev/debug work, I also recommend removing the batteries and only using them in “standalone” mode.