I don't think anyone makes an open child trailer just in case it rolls over. Burleys has a roll up shade though.
Burley and Wike are the two best brands that make singles. Both cost a lot new, but if I were to do it again, I'd go the craigslist route.

Other trailers are cheaper, and if used for a couple of years, really suffice for most needs. I'd be more concerned about the hitch. If you're going to hook it to your Piccolo, then you'd better find an axle hitch that's compatible. A Wike hitch would work and can fit a Burley head as well. For the cheaper trailers, you're better off checking them out in person.

One cool use about a folding bike/trailer combo is going to the library: I kick the kids out of the trailer and throw the Brompton inside the trailer, then stroll the bike around. (Oh yeah, for most trailers, jogging kits/stroller attachments are extra.)

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