Of course a downside to introducing EGR is, it's inert and not dense, and it's very presence in the cylinder displaces and reduces the amount of air available to burn fuel. If the space is occupied by EGR, and thus taking up more space per mass than cooler air, then you will not be able to burn as much fuel. Because of less available air. This effect will compete with the one you are trying to combat.
Of course in your case, injecting N20 just for spooling, it's not as cut and dried a picture.
Seems to me the best way you have readily available here to add heat to your air charge is to do it directly with your inter"heater".
Maybe you could also let the N20 expand out a nozzle well upstream of the intake, ahead of the inter"heater" even. That should reduce the charge cooling effect it's having significantly, no? Could adjust the control scheme to account for the delay.
TurboTR