The wax used in auto car washes is pretty weak, I doubt you'd have a problem. The biggest downside of getting the wax treatment at an auto wash (touchless or not) is that its useless and costs more. The second biggest downside is that it leaves a unpolished residue on your vehicle that will make it appear dull.
Your best bet is to hand wash at home, then polish the paint and use an acrylic wax (lasts longer) then use the touchless without wax to keep it shining between the manual efforts.
Your best bet is to hand wash at home, then polish the paint and use an acrylic wax (lasts longer) then use the touchless without wax to keep it shining between the manual efforts.