I understand your problem. I did this about 10 years ago.
I stayed at a pub at Newbridge. A quick scan on the Internet suggests it may have been the Maybush Inn, but to be honest I really can't remember, and there's a pub on both sides of the bridge. The one I stayed at was on the north bank, and I can't tell you either if they are dog friendly, although when you get to more rural areas, you tend to find they don't take the same attitude towards dogs as the townies do.
It is worth noting that the further you go beyond Oxford, the more remote the Thames Path gets, so if you were to stick your tent up the chances are no-one would notice, and you might be mistaken for an angler even if anyone did see you. I seem to remember that for a few miles after Newbridge there isn't much to see of civilisation.
A word of general warning however, in my experience Oxon farmers are probably the most walker unfriendly in Britain, so don't expect the going beyond Newbridge to be easy. At one point I had to clamber over barbed wire fences that the local landowner had put up with no thought for how walkers were going to get through. At another place between Newbridge and Lechlade, the local farmer had obviously taken the cows home along the very narrow path, leaving it churned up and in some places destroyed where the weight of the cattle had collapsed the bank into the river.
It had also been raining around that time, not heavily, but some of that stretch is very prone to flooding, so be careful and don't even think about going in winter.
I would say that your approach (Reading-source, Reading-London) is OK but it means you finish with about 4 days crossing London and the suburbs. To be honest, there's not a lot that's great about that part of the walk. Walking through Lewisham town centre (it seemed like it anyway) was far from ideal and you can't even see the Thames. Admittedly I was also hampered by the London Eye (which was on its side at the time) and the Millenium Dome which was under construction, but I did have time to peruse the Cutty Sark which you won't be able to do.
When you do get there, there's not much choice for a celebration. You can have a cuppa at the Barrier cafe (if it's open - it had just closed when I got there). I ended up walking about a mile towards Charlton (where the nearest station was) and finding the sort of pub where you hide yourself away in a corner and avoid catching the eyes of the locals - it is the East End after all. Great celebration pint I had!
Going the other way, to all intents and purposes the Lydd pool (think that's how it's spelt) is the last bit of water you'll likely see. The source itself is a mile or so further on and consists of a few stones and a monument proclaiming it to be the source of the Thames, however if there's any water running out of it, the chances are you won't get there because the river will be seriously flooded further downstream.
Anyway, enjoy it. I did.
S.