The compressor may be wearing out and drawing too much power from the engine. Or, you may have a weak electrical system and the voltage is dropping too low when the AC is engaged which can cause computer modules to go funny in the head. Your mechanic should be able to diagnose if either of these conditions exist.

If you turn on the AC, and are sitting at a stoplight..rough

I've had Hyundais and I do have Kias now - Except Niro's electric driven AC, the AC does make them run, ever-so slightly, rough.

Hyundai-Kias compressors are reliable, but they're not efficient and put lots of load on the engine,

Idle with or w/o A/C on my Ford Focus was a 0.20 Litters/Hour difference, on my Kia Stonic it's 0.60 Litters/Hour

Do you mean that the engine has normal stable reasonable RPM but that the car experiences lots of vibration?

Or do the RPM drop and the car has a low / unstable idle? Does the valve clutter change with or w/o AC?

problem remains

Check all engine/tranny/torque mounts, bad mounts amplify the vibration level of this engine when it's running the A/C.

If all mounts check out, try discounting the battery, waiting a bit and making the car re-learn its idle.

Change the fuel filter, air filter, and spark plugs - all of these clog/wear and do cause slightly worse idle.

It's worth mentioning that I have heard of Gamma GDi engines needing valve clearance adjustments even at low millage.

There are no hydraulic tappets on the G4FD, so this adjustment should be done regularly (sources claim between 40k and 60k).

ran several diffetent kinds of cleaners through

have you tried cleaning the MAF/MAP/Throttle ? those are also very likely culprits.