As a guy that uses in-ear monitors, let me tell you straight away that it's going to sound strange in ways you can't put your finger on. I drove myself crazy trying to dial my patches for me ears and I realized that it's fundamentally different. The effect of having a cab in the room is that you can feel the cab more than you think. When you completely isolate the guitar sound, now you're actually hearing the tone rather than the tone as it bounces around in the room. Do not expect to ever be able to dial it in to sound precisely like that because that's a very deep rabbit hole with a lot of frustration.
What you should be trying to dial in is a mic'd cab sound in studio monitors. What would that cab sound like if you mic'd it, then went to a control room and listened through those headphones? THAT'S the real sound of that rig and THAT'S what you'll be able to recreate in the Axe FX. To get over this issue, I did 2 things: First, I reset every patch and started from scratch with basic knobs. I didn't touch any of the advanced stuff. I even went to some of the stock patches and suddenly those sounded pretty damned good with some subtle tweaking of basic parameters. Second, I realized I would need different patches in a live band setting than I needed in the studio in order to blend in with the band members. What I use live simply didn't translate well when every instrument was isolated and really tweaked.
I would encourage you to take a deep breath, set the amp model you're using to default and tweak from a different perspective. Also, take ear breaks every 30 mins or so. Go watch TV, grab a coke from the gas station, or something. Your ear has an unbelievable ability to hone in on whatever it is you're trying to pay attention to and exaggerate it. So, if you're trying your damnedest to dial out fizz, your ear will hone in on that fizz so you can hear it clearly. Take a break, come back to it WITHOUT trying to hear the fizz and you'll be surprised with how dark the patch got while you were trimming the highs.