I must say I am a little surprised by the lack of suspension knowledge by most Stinger owners here. For having a fast sporty car, most people don't know the great benefits of larger sway bars. But have no fear, Harry is here
General rule of thumb: Slightly larger Sway Bars = A VERY GOOD thing!

Going from 15mm to 19mm is not a huge change for such a heavy car. For my old 2,400 pound Civic it was (and it was exactly the same going from 15 to 19mm) and it handled awesome after the upgrade, but for a 4,000+ pound car not so much, but it will still help button down the car and make it turn with less body roll and more composed, and I think you will all like it.
Has anyone here complained about installing an Eibach sway bar or both on their Stinger? I 'd be surprised if they did. If they did, the bar was most likely installed incorrectly, ie: without pre-loading (without doing the final tightening of the bolts with a loaded suspension). Some so called "mechanics" out there are inexperienced, so make sure all the final torquing is done with the weight of the car on the wheels! Get the work done at a shop that has a rack you can drive the car on.
PS.
Rear sway bars do not connect to the wheels as someone mentioned. In most cars they connect to the lower control arms (LCA) via the end links. The LCA's of course move vertically as the wheels go over bumps and when cornering, but are also aided by the sway bar which reduces that vertical movement. Lateral forces are also exerted on the LCA's which is overlooked by some. A bigger sway bar helps minimize some of the lateral forces on the LCA's as well. It reinforces the LCA's resulting in the reduction of lateral forces (helps the inner LCA bushing too), and vertical movement at the same time, which of course is the big one and what minimizes body roll. This all translates in better handling, ie: more tire contact with the ground, taking corners and sweepers at faster speeds, and quicker transitions, faster lane changes, etc. You should also experience less steering input which is another benefit of bigger "properly sized" sway bars.
The only time bigger sway bars are going to have an adverse effect is if you go too BIG. The car might have snap-oversteer, or more understeer (pushing/plowing), at which time you will need to upgrade other suspension components, or add a bigger or smaller bar to the front, or go with a smaller rear bar, etc. 15 to 19mm in the rear is a mild-moderate upgrade and should be nothing but beneficial to a 4k lb car, especially one like the Stinger GT that feels a little sloppy and has too much going on in the back during spirited cornering. If anything a bigger rear sway bar helps prevent rolling your tires' sidewall upon hard cornering, and makes the tires last longer and wear more evenly.