I shoot a lot left-handed, so I feel your pain when it comes to getting gass to the face when shooting suppressed.
I want to premise my response with what other posters have mentioned first prior to diving into the suppressor component of what you're asking:
AR-pattern rifles have a bunch of annoyingly finnicky parameters when it comes to making them work (excluding ammunition pressures). The lowest-cost way to solve your problem is likely going to be finding the correct buffer weight to use in the rifle that functions adequately for you, both suppressed and unsuppressed, as well with the ammunition that you choose to run through it. Your goal is to increase the time that it takes for the bolt to begin the unlocking process, so that there is more time for gasses to be expelled down and out of the barrel, thereby reducing the pressure in/near the chamber of the weapon.
If the upper is properly gassed for operation unsuppressed and has the proper buffer weight/buffer spring weight combination to function, then you won't get as much, if any gas to the face when shooting unsuppressed. Likewise, if the upper is properly gassed for operation suppressed and has the proper buffer/buffer spring combination to function suppressed, then you'll get good results when shooting suppressed. The issues arise when you try to make an upper run optimally both suppressed and unsuppressed (there is kind of a shortcut to this though - more on that below).
Think of the upper as a specially tuned and timed system. Adding a suppressor (outside of cans such as the OSS/Huxworx designs) inserts another component into the system that directly affects pressures at the gas port. If we are talking about a DI gun, this means that more hot gas at higher pressure exerts greater force on the system which, for a DI system means that there is more gas getting spewed into the gas key faster and more violently, and thereby forcing the system (if unmodified) to unlock faster. On a piston gun, this phenomenon, while not directly spewing gas into a gas key in the upper receiver (and closer to the shooter's face) means that, if unmitigated, the system will unlock faster. Basically what this means is that when using a piston gun there will likely be more gas at higher pressures present near the breech face as the cartridge ejects, but not necessarily more gas coming from the gas port towards the shooter (because piston-driven designs typically vent gasses near the piston head- there is no "gas tube" per-se). This is important in your case, as with a piston gun, you aren't trying to control the amount of gas that you are introducing into the system in order to reduce the amount of gas coming back towards the breech (and into the shooter's face), but you are rather trying to control the force exerted on the piston system so as to control the unlock time of the bolt/rearward travel of the bolt carrier group (which, technically you are trying to do with a DI system too, but the DI system has the added issue in that instead of using a piston rod to exert force on the bolt carrier group, you are using lots of propellent gasses under pressure to accomplish the same thing, with the added issue of the gasses needing to vent somewhere closer to the shooter's face-the upper receiver).
Outside of changing gas system lengths (which would mean a different barrel, and something that I wouldn't recommend doing in this case), changing the following things helps to reduce uncertainty in the system. Keep in mind that all of these components are interconnected to each other and influence the system more or less than another, and are more or less-permanent options.
Ammunition (5.56x45 NATO pressures vs. commercial .223 pressure can make a difference) (easy to assess effectiveness, less-permanent, low-moderate influence, inexpensive)
Buffer weight (moderately easy to assess effectiveness, less-permanent, moderate influence, a little bit more expensive)
Buffer spring weight (moderately easy to assess effectiveness, less-permanent, moderate-high influence, less-expensive)
Gas port size (LMT gas ports are known to be somewhat larger) (difficult to assess effectiveness, permanent, high influence, expensive, requires "gunsmithing") *I wouldn't recommend this, as making your gas port larger would likely make this problem worse*
Adjustable gas block installation (DI-only guns, as piston systems often have multiple positions that are, by default, "adjustable") (easy to assess effectiveness, semi-permanent, high influence, moderately expensive, requires "gunsmithing")
My advice would be to reduce the variables first by sticking to a certain type of ammunition, then, if you are experiencing issues still, check to see if your piston is set on the "suppressed" setting when shooting suppressed. If that still doesn't work, then increase your buffer weights to the heaviest weight where you are able to have the upper reliably cycle on a full magazine and achieve bolt lock on an empty magazine. You can stop there, or if you really want to tempt fate, you can tune your buffer weights lighter until you find the adequate balance between gas in the face, reliability, and recoil impulse. If you are still having issues, try a slightly stronger buffer spring, and then conduct the same buffer weight workup from the lightest buffer you've got to whatever gives you the right balance of reliability and less gass to the face. If that doesn't work, repeat with a slightly stronger buffer spring, and conduct the same workup with buffer weights. During all of this, check your ejection patterns for your brass: overgassed is from 12-3 O'Clock, with good ejection being from 3-4:30, and an under-gassed or short-stroking rifle being from 4:30-6 O'Clock.
Opening up a gas port would be counter-productive, so don't do this.
The "easy fix" on a properly gassed, unsuppressed rifle that hosts a suppressor, is an OSS/Huxworx can, as it pretty much doesn't affect the operating system of a rifle due to being an extremely low backpressure design. It's very expensive though by comparison to playing the ammunition/buffer weight/spring game, and won't solve the underlying issues that it seems that you're having.
You can also consider a "gas buster" charging handle. Some people like them, others can't really tell the difference, but depending on the amount of gas you're receiving in the upper receiver/outside the breach it might help.
Now for my shamelessly positive opinion on OSS/Huxworx suppressors:
I've shot the OSS cans and Surefire RC2 back to back on DI SBRs, and there is no comparison between the two in terms of backpressure. Actually, there is really no comparison between the OSS and most traditional suppressors when it comes to backpressure: the OSS/Huxworx can wins hands down, especially on DI guns. The difference is so noticeable that, for someone who hates getting gas face, it could be defined as "game changing". I've consistently experienced zero gas to the face on a properly gassed/balanced (buffer and spring combination) DI gun when shooting an OSS suppressor. Yes, zero. Here come the caveats though:
They aren't the quietest suppressors by comparison to other designs, but they aren't a slouch either. The really cool thing that I and a few friends have noticed is that the recoil impulse changes substantially compared to unsuppressed or traditional cans; the OSS smooths out the recoil impulse to something more akin to a "push" with 5.56. It's somewhat comical how much of an improvement the OSS cans make to the rifles we have used them on, even when shooting hot 5.56 through a 7.62 OSS can.
The way that the OSS suppressors attach to the muzzle device and the construction of the suppressor makes them self-tightening with every subsequent shot. This can be seen as either a feature or a bug, but it generally means that these aren't exactly "QD" in the traditional sense. That being said, other suppressor manufacturers' QD cans will carbon lock on a muzzle device if not properly cleaned/maintained, and they are a pain to remove (looking at you specifically, Surefire).
If all else fails, I'm sure that someone here would gladly offer to T&E your uppers to make sure that they function properly. In all seriousness though, you likely just need to play the buffer weight game and go to a heavier buffer, keeping in mind that the buffer that works well for your HK 556 A1 upper may be different than what works well for your LMT shovelnose.
, which is less helpful given your exiting suppressor. 
