Actually, I made a breakthrough on my own on this one, in case anybody is interested...
I found my problem was in using the word "camping" in my internet searches. After getting a near-hit, I found the term "tent-pitch" and tried that -- and that seemed to be the key!
In fact, doing that I found precisely what I was looking for in every way! In case there are other camping enthusiasts out there who are looking for unique a experience, here is what I found in the City of Bath:
http://www.campinginbath.co.uk/Really, this is a dream. Adjacent to Bath City Centre, bus service to and from, and right along the Bath end of the Bristol-Bath Bicycle Path, so you've got both those towns right at your fingertips from this campground. Best of all, if you've got a tent and a backpack, it will only cost you 5 GBP per day in the summer, 4.5 GBP the rest of the year. It is open every day of the year.
I even emailed the management to book a space for myself, and got a reply in four hours saying they will have plenty of room. They didn't have any problem with my staying there for roughly a month, either! I figured that's what I would do, since the weather will be so mild compared to what I am used to that it should be a real pleasure to camp there by that stream with my bicycle! Besides that, it's about a third the price of most hostels in the area where you have to share a room with eight or twenty other farting fellows, so I'll take the quiet and privacy, and spend the extra £££ on booze!
If you find yourself there in September, you will probably see me there!