Universal binary rewriting system
Rule 110 is a binary rewriting system that can perform universal computation, i.e., it has been proven to be universal.  It can be implemented by a finite-state transducer: it needs only finite state.

However, Rule 110 is not a tag system or a cyclic tag system, so this does not provide an instance of a specific binary tag system that is known to be universal.  It might be that examining the proof of universality of Rule 110 could yield such a system, as apparently the proof involves a reduction that goes by way of cyclic tag systems -- though personally I've never read the proof, so this is only speculation.

A side note: From Rule 110, you can construct a particular queue automaton that is universal: the queue alphabet is $\{0,1,\$\}$ and contains the state of the cellular automaton (a binary string representing the contents of each cell, followed by $\$$).  I don't know whether it'd be possible to use this to construct a specific tag system that is universal (e.g., if you can find a way to use a tag system to emulate a queue automaton).