But Andy saw it first. The FET must be a PFET and your circuit is charging thru the body diode with DS reversed so Source is on the higher voltage side
If the CSP voltage drop does not reduce to 5% of CC current, such as when an external load is on battery at the same time, it will not shut off.
5% of CC is the typical shutoff trigger threshold (+/-50%).
Sense R= 0.04 Ohms thus (for example without looking up datasheet) if V(BAT-CSP)= 50 mV which is a typical sense voltage at max current (CC) then I max is 50mV/0.04ohms=1.25 Amps and 5% of this is 62.5mA and 2.5mV. Make sure your noise is low by a careful short twisted pair of wires.
Now looking at the datasheet, I see the current sense varies with low voltage to charge slowly at first then switch to 200 mV sense voltage thus my assumption was close and the max current is 4x what I said or 1.25A*4=5A which agrees with spec.
Other
Also make sure your PCB traces are like this