In IMHO I contend you are incorrect.
As far as being tolerant - for example - they can still preach against h0mosexuality. They are NOT required to compromise the faith. For example, a Baptist Chaplain would not be required to give last rites to a Catholic GI.
Let me give you an excellent example. I met a Baptist Chaplain over in Germany. He was the only chaplain at certain small base. Since he was closed communion, I asked him how he handled communion ( most chapels do it the first Sunday of the Month). He told me that since they are so far from other bases, he thought it was nice to have a guest speaker once a month. It always happened to be the first Sunday of the month. He told me no one ever caught on!
You may be referring to Navy Chaplain Gordon Klingenschmitt. Reports stated he was court Martial for praying. That was only part of the story. He was asked to give a non-secretion prayer at a non-religious event. In spite of the instructions given by his commander, the chaplain still prayed in the name of Jesus. He was court martial for dis-obeying a direct (legal) order. He was NOT required to pray at the event. He could have graciously bowed out. In a chapel service, there are no restrictions on how he prays. I have been asked to pray at secular events, I make it clear, I only pray in the name of Jesus Christ - if the organizer objects - I will not give the invocation.
Actually being a military chaplain is an excellent choice, should the Lord call you to that ministry.
It may also be a case by case basis. As I have seen chaplains forced into doing the multi-faith crap personally and i have seen Protestants forced to accommodate Catholics and vice versa...