Reckon Cartwright would be a massive show. They'd be picking him mostly as a batsman, but one who can bowl a few overs. Has a first class average of 44, so it's not all that far fetched, even if it is a fairly short career thus far.
If they don't pick him, then we're talking Wade (who averages 30 overall in tests and 14 in international cricket since he reclaimed his test spot), Agar (first class average 26) or O'Keefe (first class average 29) batting 6. None have shown anything near what you'd expect from a test #6. Have as many blokes who average 25-30 as you want batting 6-11, if the top 5 blokes don't score enough runs then the lower order are likely to far even worse.
My inclination here is to drop Bird. I love what he's done and the variety he brings but if you're picking 2 spinners then you don't need 3 front line seamers. If things are dire then Cartwright is not the worst back up option but shouldn't be needed.
As for who that second spinner is, O'Keefe is a million miles ahead of Agar. 2 tests for Agar he took 2 wickets at 124. O'Keefe played 3 and took 10 wickets at 35. In first class cricket, Agar averages 40. O'Keefe 23. O'Keefe is also more experienced and a better bat. Agar is awfully young and there's always the hope he turns into a seriously good cricketer but test sides should be picked on the here and now.