How about having him sit on your chest or shoulder with a leash so he can bask while you walk?!?! It would look less peculiar, I guess. My Beardie usually just sits on my chest while I walk, and he seems to be very content. I have a cat tent which I use for my Beardie as well on sunny days, but I live in a secluded neighborhood where the people are very nice and respectful. I understand your concern about your shady neighbors because I used to live in a shady area too. The good thing was I had a small deck/balcony so I would let him bask in his glass tank there so people wouldn't bother me.
This is what I use. This tent is very light and portable so you could take it to the park if you wanted:
http://www.amazon.com/gp/aw/d/B002KWHE1E/ref=mp_s_a_1?qid=1368102547&sr=8-1&pi=SL75
This is what I use. This tent is very light and portable so you could take it to the park if you wanted:
http://www.amazon.com/gp/aw/d/B002KWHE1E/ref=mp_s_a_1?qid=1368102547&sr=8-1&pi=SL75