How to raise voltage output of charge controller IC due to cable loss?
I believe your problem is in those absolutely unnecessary voltage dividers R7/R8, R11/R12.
You basically adding 500 Ohm load to the charger's output and also shifting feedback voltages, messing up all the charging profiles the chip is designed to follow.

Remove R8, R12, C10 and put jumpers instead of R7, R11.

Also, what on Earth is R2 doing there? If you want a discharge path (again, absolutely unnecessary) for that tiny 10uF capacitor, put 1M there, not 22k

Finally, don't use VIAs in high current path. You have through-hole pad on VIN+. Why would you take a trace on one side only to bring it to the other with VIA if you already have same pad on the other side?

UPDATE:

One more thing. The whole premise of this post is an attempt to tweak output of the charger to accommodate the voltage loss in some (supposedly) too long or too thin charging cable. This is like attaching RV trailer to Fiat 500 and then tweaking the carburetor hoping it will go.

What if you change the cable tomorrow? What if somebody uses different gauge cable?
IMHO, you are fighting self-created problem here.

UPDATE 2:


  is there a way to compensate for the lower than expected output
  voltage?


There might be, but IMHO it is not worth doing. First, the standard way to detect end of charge of lithium battery is to monitor current while in constant voltage stage. If you use correct shunt resistor and use short appropriately sized feedback traces then battery will probably be charged OK.

Second, you should focus on minimizing voltage losses in your wiring. Heavy and as short as possible PCB traces, no VIAs, good connector, heavy and short cable. See if you can find alternative to INA219 with lower than 20uA input bias. Then you might be able to share shunt, further reducing voltage drop.

Also, I'd definitely try another chip or two to see if this lower than expected voltage is a fluke or a norm for this IC.

Regarding discharge resistor - similar ICs from TI and AD with built-in power path technology often use ideal diodes internally, to minimize both forward drop and leakage current. Maybe it worth checking those out. In any case, I would definitely use larger value. Keep in mind that it essentially drains the battery when it left connected with no power applied.