Another isssue with the mesh is that they are made from polyester, IIRC. According to the Rambam, only things which have to woven constitute a begged. For example, leather is not a begged, even when you cut it into strips and then weave the strips, because it does not have to be woven. Polyester (which is basically plastic) could be made into sheets instead of strings, so you don’t have to weave polyester. Wool, linen, cotton, etc have to be woven to make a begged.
BTW I only wear wool TK, B”N. When it gets really hot outside, like 95+, and I have to be outside for hours, then I might go with only a light shirt (no TK nor undershirt). That only happens about once a year. Going from 3 layers to one makes it much more bearable.