Thanks for sharing this. Unfortunately, no, we don't currently have a way of supporting this kind of scenario. The Dropbox Chooser page expects to be able to communicate with the original window to return the selected information and then close itself. This unfortunately fails when in this kind of single-window context. I'll send this along as a request for support for this, but I can't promise if or when that might be implemented.
Likewise, I'll also ask the team to see if we can detect this and update Dropbox.isBrowserSupported() to report it accordingly.