While it's true that MM in particular benefits from a higher ilvl weapon, there's other things you can fix that should give a larger improvement:
Chimaera Shots - 1 per 9 seconds, never delay it for any reason (except twins if waiting 1-2 seconds *tops* will let the tank finish dragging them back together after a whirlwind or shield charge), make sure you have the focus to do it the instant it's off cooldown, etc.
Kill Shots - Same deal, except it's 2 every 10 sec during execute, only thing you ever want to delay it for is a Chimaera Shot. On fights like Brackenspore you should have a ton of Kill Shots from bursting down spore shooters and mind fungus, finishing off the big add, etc. Ditto on Ko'ragh, because you can Kill Shot his shield, and on Tectus to put down the adds and then the splits.
Gems/Enchants/Scope - You're still going for crit, but multistrike is the best stat for all 3 specs now. They're basically tied for MM, so it won't matter much if that's all you play, but crit is BM's worst stat and tied for a very distant second with versatility for SV. It's not worth going back and changing them now, except for the 35 crit gems and the megawatt filament, but any new items you get should be enhanced with multi.
Potions - I don't see any used in the logs from Sat. You had 250 flask though, so that's good.
Talents - TotH goes with Lone Wolf, Steady Focus (or TotH) goes with Focusing Shot. If you switch to SF/FS, remember that you only need to shoot one FS to get the SF buff. For raiding, Iron Hawk will keep you alive better than Spirit Bond. Barrage is fine to take on every boss, but don't actually use it on a single target as it's a dps loss.
Glyphs - Aspect of the Cheetah is a must-have imo. It makes a big difference on Brackenspore (flamethrower duty), Tectus (kiting the crystals while still dps'ing), Imperator, etc. I would always glyph Deterrence too for the survivability. And unless they fixed the bug, glyphed Stampede and Lone Wolf don't mix, if you even actually use Stampede over Crows.
Spec - Here's some sims of you on a 4 minute Patchwerk fight, which is fairly accurate for Butcher and no-stands-Kargath, moderately accurate for Ko'ragh (can't account for Kill Shots on the shield, fight lasts more like 5-6 minutes) and Imperator (doesn't account for the movement or phase changes, fight lasts 11-13 minutes).
Note that Steady Focus + Lone Wolf is right at the bottom for both. The "_Multi" sims are with all your crit gems/enchants/scope swapped out for multistrike, so like I said above not really worth the gold except the really cheap stuff and the scope. Sims also undervalue TotH+FS for some reason I don't really understand other than it being partly due to movement. In the game, you may find that it's superior to SF+FS.
Last thing, I didn't bother to sim it, but I would advise switching to BM on Tectus. MM is still viable and within a few % of SV for all the other fights, but BM can double its dps on Tectus if the fight is done correctly. For spec you'd want: Steady Focus / Blink Strikes / Barrage / Focusing Shot (if you aren't extremely comfortable with the fight and planning ahead for movement, then use Adaptation). The goal of course being to keep Beast Cleaves up ~100% of the time from the moment he splits into motes.
With a competent second tank and the group that was there for N-Butcher (H-Kargath was a mess at the end, but we added Fartbox for normals), sure. Obviously no super-low ilvl people either, can't be adding hp to Butcher for no good reason.