Does he call all other women in your department the same way? If not, this man is clearly undermining your position.
On Language:
Technically, man and dude are not gender neutral, no matter how often people use it as a neutral pronoun. Is it used that way, yes, but then the secondary meaning depends on context and how close you are to a person, like how you would call grandmother, nana or, lola, or grammy. Obviously wouldn't call the unlikeable grandmother something like nana, and the more discomfort, you would probably address her as formal as you can get, grandmother.
Dude is extremely casual and unsuitable for the workplace. Sir, madame (ma'am), and miss also might be too formal depending on the type of office environment you're in. These titles were sprung from military rank and file from Western hierarchal structure and often are most commonly used when addressing a peace officer in public, a superior, or a stranger. In the office, you're most likely to use a person's name or what they choose to be called. The company's culture and standards are what you are most likely to go by. For example, in the Army you might be addressed by your last name or to be called maam, but while working in the office, the first name just might be it.
I would refer to your Human resources manual for specific boundaries, as there are no "US" standards. All of this also differs depending on the state in which you live. In California, where I live, it is against the law to discriminate against TGs.
Here's a link with all the states and their availability of protections offered:
http://transgenderlawcenter.org/equalitymapThe female version of dude, dudette is almost never used commonly anymore. Chick is probably the most acceptable and modern version of dudette.
Bonne Chance, Martine.
Audree