Wall thickness can definitely have an effect on the smoking qualities of a pipe. That being said, there are so many other factors that also have a bearing that it’s hard to quantify just how much impact it will have. The broad rule is that a thicker wall will lead to a cooler smoke, but also, obviously, means that the pipe will be heavier.
Since all factors have to be taken into account when selecting a pipe, deciding on the thickness of the wall will have to be balanced against everything else. A cool smoking pipe that is too heavy to comfortably smoke is problematic.
The rule of thumb would be to select a pipe with the thickest wall possible while still meeting the other criteria you have.