I haven't looked at Dancer 2, but Dancer 1 uses cookie-based sessions, so I feel comfortable assuming that Dancer 2 sessions are also cookie-based.

While your suggestion of using more specific cookie paths is a good one, the OP should be aware that this would provide per-backend-server (or, more precisely, per-document-subtree) sessions, not per-tab sessions. So if, for example, you had two tabs open to bigserver, then those two tabs would still share a single session, even though it would be a separate session than the one in a third tab open to littleserver.