Electronic – keep bathroom fan running for a few minutes after power off

I'm taking a guess that the B772 (PNP) collector is connected to the +12V terminal of the fan and that the circuit switches this on and off as a PWM control of speed.

To replace the NPN transistor use a P Channel MOSFET. This needs to be switched ON by grounding the gate so an inverting transistor is needed.

When the B772 is turned ON a small current flows through R1 and R2 and Q1 is turned ON pulling the gate voltage to about 0.1V. This turns the MOSFET ON this supplies current for the fan. There are lots of MOSFETs out there that would be suitable.

Be careful you don't overload the power supply of the controller

The switch is getting stuck because of the surges of current that the capacitor draws. Each time the switch closes, the capacitor looks like a short circuit until it charges up. Without the capacitor, the current was limited by the coil in the relay; with it, it is limited only by the ESR of the capacitor.

One possible way to mitigate this in the short term would be to add a resistor in series with the switch. Select a value that doesn't cause more than, say, 0.5V drop with the coil alone. This will provide an upper limit on the surge that the capacitor can draw.

A better long-term solution would be to eliminate the capacitor altogether and instead use a different kind of float switch that has a hysteresis effect. With this kind of switch, the pump turns on at one (higher) water level, but turns off at a different (lower) level. This is how most sump pumps work.

This effect can be simulated by using two float switches of the type you have, plus an auxiliary relay.