My best score in the Battle Institute is about 6,100 points (on Singles), which I believe is Master rank. Don't remember what my team was, its been a while since I've even stepped foot in the Battle Institute (and I've only battled in there like 2 or 3 times).
EDIT: Took a look at the records in the Battle Institute. The exact score is 6,124 points (Master Rank), and the team was Blaziken, Wobbuffet, and Azumarill.