I suppose I asked my question the wrong way. With the knowledge that my recruiter actively believes I am not entitled to a CAC card, what is the best way to go about getting one that does not feel like a disrespect to my recruiter? With what @CoMa had mentioned, his personnel specialist had advised him to bring orders, which I’d like to do as well.
Your recruiter is trying to prevent you from obtaining something you rate. Who is disrespecting whom in this scenario? Just get the CAC. You're in the program. Don't think too much about it.