No one cares about Stack badges except stack users. If your clients have never heard of Stack Exchange, as with most people, then the badges only serve to confuse and distract. They offer no added value.
In addition, having a "badge" doesn't make you automatically an authority on the subject. Badges only reflect your interaction on Stack sites, not your overall knowledge and expertise. It is entirely possible to be hugely knowledgeable on a subject but not have a SE rep which reflects your knowledge. Just as it's possible to have a good deal of SE rep, while still not actually being an expert on the topic. SE badges only matter on SE sites, clients really don't care about them.
Leave them all off.