Look at it this way. Talking face to face with someone or on the phone is one thing, however, this woman is typing it in a chat. Speaking may just be a natural thing but you have to think what you're typing and you're choosing to type "sweetie". And again, most times if you're talking to someone and you correct them they understand and apologize. Again, I'm not big on cutesy names like this, and yes, I'm in and from the south. Born and raised. But on your job you expect people to be a little more professional.
Doesn't matter whether it is a chat or not - a chat is different than other forms or written correspondence - a chat is just that chatting (informal) so it is implied that you would write as you would speak. It is similar to a phone call but in written word.
And no plenty of people do not think when they type -
I think the big difference here is what YOU expect -- which can be very different than what another person expects. You seem to have difficulty understanding that people have different expectations. Not everyone has the same expectations as you do even in a professional environment. That is why you kindly clarify how you would like to be addressed.
To answer your simple question (and most seem to see it the same way)
Is this condescending - NO. How would you approach - clarify that you would like to be called Barry. We all agree on that - how best to address - difficult to say -- humor by turning around calling her tweety bird after she writes honey - or being kind and firm - please call me Barry each time after she writes honey or whatever the heck she does.