Can an INA219 current sensor operate with a FET body diode present over the sensing pins?
It looks good to me too:

The turned off mosfet will basically disappear from the sense circuit and all current will flow through the resistor instead, so everything should work just fine.

I initially thought (before I read your whole question) that you intended to use the internal resistance of the mosfet as the current sense resistor and were concerned about possible damage to the INA219.  but that would also work.