The 5 volts is coming from a AMS 1117, which has short circuit protection. But a short means the supply device turns off, and then the regulator thermally cycles, shutting down at 165 degrees and then restarting. That can't be good. But anyway, the unit switches off when there is a short, so the current must be limited.
A polyfuse means the system will still go off.
I will look at the FET option, but I don't know much about it. how does the circuit work?
Also, I know this might not be the usual way, but what if I sized the limiting resistor such that half the voltage was dropped across it and used a 40mA charge pump in the receiving device. I don't think the current for the uC and opamp circuit varies much. If the uC and opamp circuit is drawing 40mA, then can I assume the input resistance is 125 ohms?
So, I size the limiting resistor at 125 ohms/ 250mW and use a 40mA charge pump in the receiving device. Unorthodox, but ....?