Geek gal, I just went through almost the exact same decision, right down to already having one of those rear strap-on racks.

The only things i can add are these:

The weight of a hitch receiver, or even the receiver plus bike rack plus bike, will not make a detectible difference in fuel economy. And the rear is certainly the best exterior place by far to put the bikes in terms of ease of access and fuel economy. Any fuel economy hit from wind resistance is likely small, and zero when no rack is used.

I tend to keep my cars forever, and I cycle a fair amount. I've coped with carrying bikes inside vehicles (bad idea!) and with fighting with the strap-on carriers and watching them constantly through the back window. So I put on a hitch receiver, bought a bike carrier for it, and am happy with it all.

If you are going to keep your FEH a long time and bike enough for the cost of the hitch and carrier to be worth it, by all means do that!