Hi,
"I think I'll remove the diode from the Charge path and put it before M3 Source pin." - Best not remove that one, both diodes need to be in their respective places.
If yi-yun placed the word "language" in an international language or a picture of a flag (so that every human can guess what it's for) on their website, they may get more English-speaking traffic rather than landers and leavers... I looked at their site for a technical document but the language barrier makes that impossible, and other online marketplaces do not provide anything beyond sales-level overview technical details. Thanks for posting the image of the part, anyway.
R.E. last point: Not sure, maybe adding another boost converter might help but I think a gate driver IC (as suggested by Klaus higher up in this thread) will make life far less arduous for you to deal with level-shifting issues if they arise.
If you can get this to work using relays and omitting semiconductors which appear to be adding a lot of unexpected issues, I strongly recommend it.