alex-sh, I suspect your Ebay module will work fine. It's a combination of the TP4056 charger and B62860 boost converter, which is the same as the MT3608. I used exactly that combination to get 9V, but using two separate modules. No reason why they can't be on the same board.
But I do wonder what happens when USB is plugged in as far as isolating the battery charging from everything else. Ideally there should be a load sharing circuit to let USB directly power the boost converter so battery charging can proceed indepently to normal shutdown. But you can test for that when your module arrives.
Thanks for finding the module.
Nusa, yes they pretend it's 9V because it's in that standard package with 9V battery connectors. And for a lot of uses, it should work ok. By the way, did you see my video on multiplexing by segment? You and ogden got credit at the end.
Now if I could just understand what balanced charging is.