A couple of things come to mind. One is that old TV cigarette add: "What do you want- good grammar
or good taste?" It sounds like you get the job done without the sugar coating. Personally (and professionally) I'd rather work with a no nonsense straight from the hip shooter like yourself than some bleeding heart do gooder.
So my advice to you is to learn some acting skills. If empathy is what they want, and it's not a natural part of your personality, then learn some lines. And always be sure to say the person's name you're addressing:
"I'm sorry you feel that way, John. Is there any sort of arrangement or compromise we can come to that would satisfy your needs,?"
"I'm doing my best to understand what you're trying to say, Jane. Perhaps if you rephrased your statement, I could respond more appropriately."
"Tell me how you feel about that, Judas. Then we'll go from there."
The lady who started Mary Kay (I guess her name was Mary Kay) said that she became successful after she realized that her job was to make Clients FEEL important. Now, I'm relatively sure she was no Mother Theresa. But she had the ability to give comfort through her products and make a profit to boot!
I guess that's what you have to do. If your people are looking for empathy, you need to give them a reasonable facsimile. As long as you know you're being fake, it's okay. To thine KNOWN self be true.
The best to you, desertnurz.