Since there are already many great answers covering different aspects of the question, I will just focus mine on what could be your first steps with the BM for him to adjust his attitude toward this issue.
First, bring the issue to the BM's attention. The BM seems to be a very valuable resource, very invested in his work. If he is aware that the issue is important and that you expect him to tackle it, there is a chance that he will adjust.
I would address the issue as a high turnover issue and NOT as a employee low satisfaction one. It's a much less ambiguous metric (he can argue that reviews are biased/irrational, but he can't deny that employee are leaving), it's easy to measure, it has a cost that can be estimated, and it doesn't make things personal. You are not telling him that he is the issue, you just ask him to solve the problem. He is in control to investigate the cause and act accordingly.
Second, give him data and resource to investigate. Employee satisfaction survey is one of them. You could also produce an internal report about the cost of turnover, knowledge lost, time for new employees to be operational, hiring cost, and other costs induced by the departure and arrival of employees.
Finally, let him come back to you with a plan of action and check periodically the results with him. He may adjust, or he might find another solution you didn't think of (who knows, maybe he won't adjust as much as you wanted but he will find a way to attract employees looking for jobs with a high level of commitment and you will end up with a compromise that answers your worries).
And if he can't solve the problem, you will be both facing the fact that he is currently unable to solve this, and whatever the course of action you would like to take, he will probably be more inclined to agree and go along with your decision.
That's just my thoughts with the context you gave.