FWIW, I’ve never played through a consumer grade FRFR cab that sounded “right.” They’re normally boxy in the mids or have boosted top end. When I listen to a real mic’d cab through my studio monitors and A/B it with an IR, it’s identical, so I think it comes down to the color the FRFR cab adds to your tone.
If your studio monitors aren’t sounding good either, I imagine you’re getting a lot of reflections in your room that can enhance harsh top end. I HIGHLY recommend investing in some room treatment to smooth everything out. Your listening environment plays a massive part in your overall experience.
Since Meyer cabs are pricy, here’s a potential solution that might help you with what you already have that will keep your direct line sounding good (what you hear in your headphones) while making your FRFR cab sound better.
You’ll want to route your FRFR cab to Output 2. Try creating a preset that’s just Input 1 going to Output 2. Connect your phone or run a line out from your DAW into the Axe’s Input 1 so you can play music (something well-recorded that you’re familiar with) through your FRFR cab.
On your Axe, go to Setup - Global Settings and page over to Output 2 EQ. While listening at a decent volume, make EQ adjustments to get the music sounding balanced and neutral. You make need to cut mids and/or highs and boost low end. Once it’s sounding good, raise the volume and see if you need to make further adjustments. This is basically “tuning” your FRFR cab the same way a FOH/Monitor guy tunes a PA.
Now that it’s sounding good, try one of your presets and see how it translates.
As others have mentioned, a 1x12 cab isn’t going to hit the same way a 4x12 will, so you’re listening to it more like a floor monitor giving you the mic’d signal rather than hearing a roaring 4x12 behind you.
I hope this helps, and sorry for the long post.