I have the Eibach bars front and rear. I think they made an improvement, but not nearly as much as coilovers. Sway bars were my final suspension mod. All my driving is on the street. I have most of the supporting mods as well (Ground Control coilovers 440 lb/in front 650 rear linear springs with custom valved Koni Yellow dampers, M3 arms front and rear, subframe bushings, front tension strut and rear LCA poly bushings). I have had an LSD since I bought the car.
I would say the "proper" procedure is tune your springs and dampers to your liking first, then asses if the car has too much roll or has understeer or oversteer characteristics. If the whole car rolls too much, stiffen both bars. If the car understeers, stiffen the rear bar. If the car oversteers, stiffen the front bar. If you are going to get an LSD do that before the sway bars because it can affect understeer/oversteer characteristics.
Honestly, I would save your money for an LSD (Quaife/Wavetrac for the street) and shocks and springs first. If that is out of the question, asses the roll stiffness/understeer/oversteer characteristics. Have you done the subframe bushings yet? If you have higher mileage (especially 100,000+) you may just benefit from new shocks. I like my Koni Yellows. I have had them since 2013. You don't want to add a new sway bar if your shocks are worn out.
Which M3 arms do you have specifically? There is the front lower control arm and tension strut, the rear upper control arms and the rear lower control arm. If adding a rear M LCA to a non-M subframe, you will need a rear
adjustable toe link to get toe correct (at least I did). The M3 rear LCAs are longer to increase camber, but the M3 toe arm and subframe are also different. I was unable to get proper rear toe with the non-M toe link and subframe.