I would have said that it is equivalent to connecting a load to the inverter's output itself but not modifying the output impedance of the inverter stage.
If I were you I would have tried the topology seen below:
View attachment 197685
This is exactly what you want to achieve (as far as I understood). The output can swing very close to the rails, if the input from the previous stage has increased, the output inverter-based stage will push/pull current into the load while keeping the gates of the devices( their CM) constant ish. That ensures that the output can swing close to the rail before transistors in the output stage enter triode (especially the active one).
Of coarse you have to make devices in the output stage wider to drive a really heavy load (low-impedance one).