Half Hot Outlet Tripping when Light is connected
You said that the tab on the hot side is broken and from that, I assume the tab on the neutral side is not broken. The problem is that the same neutral is being used for both the switched circuit and the load side of the GFCI.
A GFCI works by comparing the current through the hot and neutral wires. If the current is unequal, it assumes that current is escaping through the ground and trips. In your case, current from the switched circuit is returning through the GFCI. Since this current did not originate from the GFCI, it trips.
Because there isn’t a separate neutral that bypasses the GFCI, you cannot make this configuration work. Your choices are to either (1) abandon the switch and make the outlet always-hot or (2) put GFCI outlets where you want protection and not use the load sides; instead, use pigtail wires so that you can connect each outlet’s line side to the circuit. The half-hot outlet cannot have GFCI protection in this case.
The other option is to use a GFCI breaker to feed everything and restore the GFCI outlet to a normal outlet.