I once made a 60W LED driver (2.8A output) with undersized current shunts. I took a factor of 2 with the power dissipated v.s. maximum. The PCB itself was getting very hot (up to 65C) because of a small form factor (believe 45x60mm) and many SMT cooled components. I believe the board had to 'cool' 10W - 15W from all the components. I turned it on. Initial current about 2.7A. After 1 minute it was 2.85A. Turned it off, grabbed the board, burned my fingers. Ouch!
Great.. I think I need a second fan to cool the electronics

10W sounds a lot, but it was a buck-boost converter which doesn't have ultra high efficiencies. (not the converter I was using, anyway).
Good job on overspeccing your current shunt. A 5W resistor may be fine too, but anything that helps make it even cooler is good. Still, avoid it being warmed up by MOSFET's in the area, if you dissipate 5W in a MOSFET on the same PCB or heatsink, it will still drift..
Your current sense:
4.096A @ 20mohm = 81.92mV. To get 4.096V, amplify by 50x.
According to the datasheet, the INA138 outputs 200uA/V. The input is 81.92mV, the output will be 16.384uA. To get 4.096V it's simply R=U/I=4.096/16.384u = 25k. Parallel up 4x 100k's or something.
The tolerance depends on your resistor initial accuracy, it's thermal drift and temperature, the gain error of the current sense amp and your output resistors. They pretty much all add up, so a 1% shunt, +/-2% sense amp and 1% output resistors is a total of 4%. Excluding temperature drift, which can easily add 0.5% if you push the limits of a horrible resistor.
For e.g. a 100ppm/C resistor heating up 40C is 40*100ppm = 4000ppm -> 0.4% drift
A 300ppm/C resistor heating up 70C -> 2.1% drift
If you want to use the max chip, you can divide by 1.2 indeed. 15k & 75k should do that.