When you say you set the controls the exact same way JP does, what precisely were your settings?
In particular, what were the dB settings on the 5 band graphic eq? If you set these visually by setting the relative distances of the virtual pots to match that of photos of JPs amp that could be the issue. The taper on the real JP2C is very logarithmic, and you get almost no change until you get to the ends of the pots then a rapid change in sound over a short distance. The Axe eq is linear.
On the mark amps, the amp controls set the feel and character of the distortion and the eq shapes the overall sound. My understanding is that the amp controls in the Axe are spot on for the JP2C (and probably all the amps), but again the taper is different for the graphic eq. So just diming the amp mid control won’t do what you want.
I have found the JP2C to have loads of bass and mids. I often set the mids to no more than 2 for rhythm and 4 for leads. The 750hz I set at -5 for rhythm and -4 for leads. From left to right I set the eq controls at 3, 2, -5, 2, 1.8. I do these same settings for all the mark series and Triaxis as after lots of experimenting these sound the best to me, even with lots of different cabs. They seem to accord pretty well with JPs claimed settings (at least for the mids) which can be found in the video description text under this video:
Also what IR were you using? I keep coming back to the ML USA TRAD in Mikko’s Greatest Hits Pack. Amazing. YA Recto ST which come with the Axe are also great (mixing SM7, 421 and 121 cabs sound great at equal mix - try the DRK version).
Disclaimer - I am a bedroom player and haven’t tried this live but I think the logic at least holds up.