Found this post when searching for ways of upping the discharge current of the DW01A. I thought the TP4056 chip was only used for charging and its the DW01A on the better boards that handles the over/under voltage and over current protection. (Note I said better boards since some don't have the DW01A).
One of the better description of these modules is TP4056 / TC4056A Lithium Battery Charger and Protection Module which includes handy links to the datasheets for both main chips and even the protection mosfets.
So, really the charging current is immaterial to the discussion of how to drive higher current devices using the TP4056 module. Its the DW01A that needs to be tweaked. According to the datasheet, it has a typical 1.5A and a 3A max. it appears that this current is not running thru the chip itself, but thru a pair of mosfets that act as switches to basically cut power to the cell when protecting. The chip also has a current sensing pin.
So, one approach is to mod the current sense and replace the protection mosfets with a pair that will handle higher current. The dual N-channel in a TSSOP-8 package isn't going to handle big loads and only has a 1W power dissipation (even though its rated up to 3A to 4A).
Your other problem is going to be the MT3608 which is only rated at 2A and without a heatsink or proper cooling, I doubt it could handle 2A continuous. Why you need to bump the 3.6v to 4.2v of a lithium to power 1W LEDs is beyond me. You know they don't last near as long when over driven not to mention the worry of heat dissipation. I've gone in the opposite direction and gone with 10W units under driven to stay cool.
My recommendation, if possible, would be to try and divide and conquer by run multiple TP4056 modules in parallel. And if you are pulling 3A to 5A, what kind of single battery is going to last long at that kind of draw? So, two modules, two batteries feeding 5 1W LED each might be the ticket.
My project is trying to run multiple 100W LEDs somewhat under driven without draining battery power in a hurry. My solution is to use POV, aka persistence of vision, by using a 555 timer driven decade/ring counter thru power mosfets to rapidly switch the power to only one 100W LED at a time. Do this fast enough, and they all appear to be lit at the same time, but your current draw looks like only one is on at a time.
Then on the other positive or negative "leg" of each big LED is also connected to another MOSFET with is then PWM dimmed via Arduino. So, the POV ring counter is refreshing 100x if not 1000x faster than the PWM. I'm using both a N channel and a P channel MOSFET for this. You might really save on power by doing something similar with your 10 1W LEDs, but at the cost of more components and complexity.