What Verbal said....................
If I'm in uniform, and I am offered a discount, I accept gratefully and I appreciate it.
If I'm in uniform, and not offered a discount, I pay what they say and move on. There's no "WTF ? Where's my discount ?!!?"
Do I expect a discount ? NO. Do I appreciate the occasional courtesy ? Yes. When I accept one, do I feel like (as you do) that I am "abusing my job title" ? No, I don't.
When I signed up for this job, my overriding thought wasn't "Man, I am sooooo gonna rack up on that sweet 10% off at Captain D's !"
It's your family business. You don't want to offer a LE/Fire/EMS/Military discount ? Then don't. It's not a tough decision.