I have redesigned the board. I added a momentary tactile switch between GND and CS pin, to reset the DW01 IC after a undervoltage lockout. In addition I have added two battery connectors, one JST PH that will connect to many Li-Po batteries, and a 2p header at 2.54mm pitch. Finally I made a hole in the PCB for pushing out the battery from bottom.

The boards were ordered from JLCPCB. 

The components I ordered from LCSC.com and tested DW01+G and DW01A-G. According to the datasheet the last one should have auto recovery, which means that after a overdischarge the chip goes back into normal operation when the battery voltage exceeds 3.0V. I expected to measure 0.1 or 1.5uA shut down current (when vbat is <2.4V) but I measured 20uA current and when the load was connected I measured 2400uA (exactly 2.4V/1kohm). Then I found I made a silly mistake in the design

I wired the tactile switch such that it is always connected. Then I removed the switch completely and continued testing. I noticed with DW01+G the circuit was working fine, it was low power, and it does not have auto recovery as expected, but it recovered after the tactile switch was pressed, which you may forget. The DW01A-G should have auto recovery but it didn't work and I don't understand why.  Only after pressing the tactile switch it recovered, but this is not very handy. I want this circuit to automatically work and not require manual action. 

Then I remembered I still had a bunch of $0.15 18650 3A protection boards, like this one. 

I hooked it up 

And then it worked as expected, after a overdischarge (<2.4V) it disconnected the load and battery voltage could rise again untill it exceeds the re-enable voltage and the protection IC was back in normal mode . I measured 2.3uA in normal operation and 0.9uA in overdischarge state which is great. Zooming into the chips shows that it actually uses the DW01-A

Ok so now the circuit is working as expected and I need to design a new PCB using above protection board. 

I designed V4 of the board with the battery protection boards as a module and immediately ordered from JLCPCB.