Luke, I absolutely agree with you regarding "speculation posts" from any source. I also totally agree a picture is worth a thousand words.
I've been searching for some photos that might be helpful and found one, a painting of Brg. General Hayes (organized the unit) that is interesting in that it shows the peaked cap as you mention, and the other, the only one I've ever seen of the ranks in the side cap. You can see the eagle badge and it looks to me to be worn pretty high up, maybe to allow for he screw post???
I will continue to keep looking as you never know what might turn up.
As for how many tech reps there wee and/or stationed in the UK I have no idea. If I find anything in that respect I will share it.
Terry