Yes, the FRFR would only be used for your personal monitoring. You could use it as a floor wedge or as backline, but in either case, FOH will be doing all the heavy lifting, and is what the audience hears.
People get good results going direct to FOH, but in order to get there, you have to use presets that translate well to your system. The stock presets are a good starting point, but try to create your own. I make my presets on a CLR and have FOH in mind, so I'm not particularly after the "cab in the room" feeling. Whatever preset you create on a CLR at gig volume should translate quite accurately to what FOH hears.
Your friend's "fullness" is probably the cab in the room. Plus, his 4x12 is probably coupling with the floor, adding to the sensation. What the audience hears is a miked cab sound, not the "cab in the room" sound (assuming his cab is miked, of course). And guess what--a miked cab sound is what the Axe FX excels at. So as far as FOH is concerned, you are both on a level playing field, so what really matters is to make presets that translate well. In order to do that, it helps to do it on a monitor that faithfully reproduces what you feed it, and do it at or near gig volume. When I first started transitioning to FRFR, all I had were near field monitors that couldn't get very loud. I thought I had made some awesome presets, until I finally tried them out on some CLRs, and discovered the presets were just gross at higher volumes.