Unfortunately, AA may be the best bet from AUS if you want to limit connections and overall travel time. We fly from SAT with one stop in MIA on AA leaving at 06:20 and arriving SXM at 14:33. Connection is somewhat tight with 1hr12mins in MIA. Looks like AUS has a similar connection leaving at 06:30. Check google flights. The straight morning shot to MIA is not always scheduled for everyday of the week.