I think EAP would be a great thing to try! All the same privacy laws cover EAP programs. It would still be a violation of HIPAA if they told anyone, and HIPAA violations are pretty serious matters. My guess/hope is that the EAP counselors are likely very familiar with PTSD in officers and would be able to help.
I can see how administration might be jumpy about PTSD now. Basically, someone decided to tell them that they had to stop working there because the job harmed him. Now administration, the people who give that job out to begin with, they are putting down anyone who says the job is harmful and can lead to the injury of PTSD. It smells like a lot of denial on their part. That's THEIR stuff.
I think you are smart to see it and recognize it for what it is, and to be careful around them because of it. But, they are kind of full of it in terms of what is and isn't PTSD, and what it means about the people with it. So try not to take on their stuff as your attitude toward your own PTSD symptoms. (Which is so super easy to say, but hard for most people to do.) Just the fact that you are considering therapy makes you are braver than any of the folks in administration with that attitude.
I think it's really normal to reconsider the job, especially when having active PTSD symptoms related to the job. Some of my PTSD symptoms come from my old job. Don't beat yourself up for it. PTSD is awful. It might be time to quit and move on, it might be time to stay. If you get into treatment and keep your much deserved recovery from PTSD first, I think figuring out work will become a lot more clear. Everything that you are experiencing can get so much better.