I work in a large financial services company. I wear a ring just to the office and shake hands with women in business settings. IMHO, when there's reliable/sufficient/daas al mi lismoch room to be meikel, I try to be as normal as possible. It's hard enough never being able to eat the same food, having to run out at 3pm on a Friday, and taking off Thurs-Fri for 3/4 weeks in a month (Tishrei). ESPECIALLY when outside of NYC/large cities, this may be the only time the woman meets a frum jew. I'd rather rely on the kulos than have her feel slighted (with a quizzical explanation in her mind) during the only interaction she'll ever have with a frum jew.
I want to ask you a very specific question here:
What would happen if you (politely) didn't shake hands? Would you get fired? Would the client not follow through with the transaction?
From your statement, it appears that your self-heter to shake hands is coming from being image conscious, as you believe that they would see it as "awkward". Putting self-consciousness before Yiddishkeit goes against everything that Judaism stands for. Open a Shulchan Aruch and read the first 10 lines.
It's odd that you want to give off a certain impression of Frum Jews, and you are shaking hands with a woman in order to do that. Think about this.
I'm not here to give you mussar, I'm just reflecting on your statement.