discharge power rail with MOSFET
Since you are actively driving the enable to either 0 or 5V, and since the control signal is always valid, even before 3.8V is enabled, you can just use PMOS with a current limiting resistor. I would select the resistor so that the RC is around 20 ms. This way, after 200 ms, you will go through 10 time constants. The rail will be pretty low. Here is the circuit.







simulate this circuit &ndash; Schematic created using CircuitLab

One problem with this is that it will only drain until it reaches the Vgs(th) of the transistor. If that is not desirable, you can use your original idea of pulling it down with NMOS. You would need to invert the signal. Make sure you drive the NMOS gate with 5V so that is is on hard regardless of the voltage decay of the 3.8V rail.

Anyway, if you use the above, you can probably use any power rating resistor you want, assuming the 3.8V rail is de-activated, and the FET will not be turned on in a high cycle fashion. If you do need to cycle rapidly, use a real power FET and a 0.5W resistor.