I have this problem in general, with tone knob, EQ, and I have fine dynamics. I just find the harmonics of these strings and their tone a bit shrill at times and if I roll off the highs, I end up with muddy and round Low E through G tonality. Although I find this issue is often on all my guitars, I'm more specifically focused on running into this with the main model I play, which is a 57 Les Paul Junior with a dog ear P90.
Possibly preaching to the converted, but I have been using LP Juniors for a while now, and the issues you report feel like home ground.
I have LP Juniors from '55, '57 and '59, also a '57 LP Special, and a couple of P90 equipped Gold Tops (one reissue, one "uber" replica, the "student" models are all original). Obviously brass saddles aren't an option, as there aren't any saddles, but almost all my Juniors/Special now have MojoAxe wrap-over bridges. The improvement in intonation is clear, but kind of peripheral for me. The big deal is that the MojoAxe bridges throw off fewer harmonics, especially when playing on the lower frets with high E and B strings. Not cheap, but definitely good value for me.
Weirdly, for most amp models, I tend to control the rest of it with the tone and presence controls. The low E string, close to open, can give any amp or model a hard time, but rolling off the bass helps. I tend to try to control the highs as much with Presence as much as with Treble in the amp models, and the tone controls on the guitars get used in a way I'd never consider with humbucker equipped guitars or more "normal" single coils. I sometimes roll them back a long way further.
Liam
PS. I know you are not looking for guitar based solutions, but my favourite, the '55 LP Junior, ended up with '59 size frets in it a good few years ago. Massive mistake! A very lively guitar that ended up trying to play itself. I got it re-fretted back to exactly original '55 size wire, and immediately returned to the guitar I knew and loved. Weird to discover that some of the intricacies are more about the guitar than the player!