It seems obvious to me - 10 Barnes, 12 Carter, 13 Horne.
Barnes is likely to provide our best attacking spark and tactical kicking (God, can't believe I am in any way endorsing Tahs try tactical kicking!!) and robust defence at 10.
Carter for all the reasons Bruce enunciated in his excellent post. I don't give a flying fuck if he's good enough for the Wallabies or not, he's good enough for Super rugby; he has done the job, and is really not the problem with the Tahs.
Horne seems the best 13 option, and maybe Carter can get and keep him aligned better in defence. Horne can tackle, but goes for too many big hits and rushes up too much, so misses crucial ones. Good tackler in general, bad defender at the moment.
Bench Hangers, and if he needs to come on Barnes can go to 12, Carter to 13 if needed, Foley can move around, there are options.
BUT - this is predicated upon the assumption Barnes can kick goals.