My fault... Who else?

I discovered the issue: I had soldered the DW01A protection IC rotated by 180°.

After replacing the chip—and swapping the 8205 dual MOSFET as well, just to be safe—the battery charger now works exactly as expected.

The controller can correctly read the charging status: charging, full, and discharging all work. Only the “battery not connected” state is still not detected properly.

Unfortunately, the TP4056 attempts to charge even when no battery is connected, which causes the status pins to toggle continuously. I had hoped they would simply stay low in this condition. I definitely don’t want to detect a toggeling status pin....

The battery voltage is now correctly measured and advertised in the web interface.

By the way: resoldering the barometer with hot air also solved for the short on the lines. The barometer is now working perfectly!