The most important part of project management is leadership, business acumen, and execution. In other words, project management is simply about management. You have a varied and extensive background to be successful, so it is just to learn the basic tools, techniques, and methodologies about professional project management that may be of interest to round out your experience and help you be more effective.
PMI and PMP is always a good route, but that depends on if project management is the field you want to be in, or is it the discipline that you want to add to your portfolio of professional education and knowledge. Perhaps a certificate and classes are best.