As a result, I'm probably going to go down the pannier bag route and spend around £100 to £150 buying a half decent set.
Any recommendations in that price range?
If you're willing to spend that much, Carradice and Ortleib are the ones that come to my mind.
In the mid-range, I'm quite happy with my Basil panniers. My most-used ones are Mara XLs, but they are very definitely not quick-release which upsets some. They do fold/roll up when not in use, which minimises aero penalty (important on the open fens), although there is no clip for this probably-unintended feature, so I use bulldog clips or pegs to hold them closed. Each side takes a US/EU-style long-handled cloth grocery bag dropped in, with a little space for a lock or toolkit in front, but I think Ortleibs are a similar size/shape.
Which reminds me: why is having your repair kit under your clothes a problem? When you need the kit, you've not yet started work, so your hands should be clean... and once you've done the repair, clean your hands on the cloth from the kit using the cleaner in it (I have blue gloves in there too, but I often tear them!), so your hands should be clean when you close it and put it back. If the kit gets crap on the outside, wipe it off with the cleaner... maybe keep an old plastic bag to drop it in if it gets too bad. My repair kit usually lives in the bottom of my saddlebag (which gets thrown in the pannier if I've got the panniers attached and there's space - it's faster, which is especially important in winter).
Are
Tredz a reliable retailer?
No: they're now
Halfords and are like the shops but worse, with a long history of tricks like showing stock they don't have and stringing people along, even before they were bought. Search this site and I'm sure you'll find some reviews.