The “business handshake” is not a “wing” issue.
For instance, there are many people who are quite frum and not meikilim in halacha but who hold it’s okay to shake hands with a woman in a business context and the like, albeit (in my understanding) with provisions. Those are, say, if a woman extends her hand to shake yours AND it would cause her pain/shame if you, even very nicely, explain that you don’t shake hands with the opposite gender.
Ask your LOR, but this is not a “wing” issue.