My car also has the only grille below the bumper. It is totally blocked now. Mine also had an air dam of sorts underneath just below the front of the radiator. I took it off. In addition to that, I now have a belly pan under the engine compartment all the way forward. I have no heating problems but just watch yours till you're sure. A grille block sealed properly is one of the sure fire methods for FE. You have a PM from me.
the air dam scoops up air and cools your radiator. keep an eye on your temp. guage if you block it off. I would cover 1/3 and check during a drive, then 2/3 and recheck, finally block it all and keep checking temp. you might be OK as long as you don't overheat
My in-bumper grill is totally blocked off, I only saw my temp gauge go up two little notches when I did this and noticed a quieter and seemingly smoother ride. I have yet to do a coast-down test but it definitely improved my aerodynamics. I used a sheet of black coroplast attached with "industrial strength" velcro, this way it's easily removable and modifiable.