The Secretary of State (or local county clerk) is responsible only for ensuring that names are not identical -- and "Flutter" is not identical to "Salon Flutter."
There are a couple of ways you can approach this. First, you can consult with an attorney to see if there are any state trademark bases for getting the other company to change its name.
Second, you'll want to find out from customers whether there's been any confusion about who's handling eyelash extensions.
Finally, from a marketing perspective, it may be worth exploring with them whether they really intend to offer the full range of services you do. If not, you may find that the company you thought was a competitor could become a strategic ally.