He might have had a clause in his employment contract about using his official position in ways not in keeping with policy, etc., etc. But if not, he might have a case.
He didn't use his official position for anything. He wrote the letter as a citizen, not as a spokesman of the department. He isn't identified as a Lieutenant within the letter, or claim his department supports his beliefs. The title that appears under his name could have been added by the paper, and even if he wrote his title under his name when he sent the letter in, he did nothing wrong.
You are correct I just saw that he signed the letter with rank, and employer. That’s a no no in almost all circumstances.
No matter what else you never speak for your employer. That’s why they pay people to do it right.