Surgtech, if at all possible, I would suggest taking a class in some sort of management. It would be great if that course could count towards a bachelors degree down the road, but that may not be necessary. Something to show that you are striving to continually improve your skills. Perhaps a certificate in management or something. There are online courses as well as community colleges that offer such classes at reasonable rates. Your employer may even have a tuition reimbursement program. This might not help with the current position you are seeking, but will put you in a better position for the future.
It's better to be proactive than reacting when something comes up. Be ready for it when it does.
That being said, I don't know your particular situation at work. But I've found that even with lots of experience, if the position calls for a certain level of education, it's difficult to get that position without that education (even if you do the job already!)
Even though I have two masters degrees, I still go for training and certification in specialized areas. I recently took a 3-day course that led to me being a Qualified Assisted Housing Manager. I develop affordable housing, and this certification not only taught me some things, but illustrates to my clients that I understand the management side as well as the development side (although I don't directly do the management side).
Now I'm going for a specialized environmental/sustainable design certification to put myself in a better position for future projects. It'll also help my networking.