Hi.
I recently bought TP4056 modules with additional protection thinking It will solve my problems but i fell into a rabbit hole.
What I'm doing is a LED lamp that should be battery powered and chargeable. Sure, you can just switch out batteries but where's the fun in that? Also what if you want it on for longer?
And I thought I can just use a TP4056 and it claimed an overdischarge protection, which I wanted, because I want to gift it to someone. I want it to be foolproof.
So this is What i came up with. 1 is usb power, 2 is battery power, 0 is off. Ok It works.

Then I tested it and it ran until battery was 2.6V when I turned it off. So it doesn't work? Well it has a DW01A chip which should trigger at 2.4V overdischarge. I don't think that's good. So I started reading.
I came across this
https://www.best-microcontroller-projects.com/tp4056.html and a few similar topics and after reading through a lot of stuff I'm not sure what to do.
I also noticed the module doesn't work with USB C-to-C chargers because it doesn't have CC lines resistors and no way to add them.
So I thought about spinning my own modules from PCB supplier but with some changes.
1. Add 2 5k1 resistors on CC lines so it can be used on any USB port.
2. replace DW01A with FS312F-G which should trigger overdischarge at 2.9V
3. Add a mosfet switch instead of manual switch for usb connection
Please do comment on all 3 points, but I'd like to know how to implement the switch.
Should this work?

Is this good enough? The source just mentions schottky diode can be replaced by a mosfet but it would need to be tied into a sensing circuit.
It's been a long time since highschool when I learned about this stuff. Wouldn't a mosfet have less dissipation and voltage drop than a schottky? Surely if one can be closed on usb voltage, I can do the opposite on schottky line, close when no voltage on usb line. So I could use a second 2-mosfet IC? P or N channel? How?