The first thing that has to be done (in court, or via lawyer-to-lawyer communication) is that The Company has to prove that they own the copyright. If they accomplish that, you can defend yourself by providing proof of a license to download and redistribute. From what I can tell, you cannot directly prove that, since the rights-holder did not give you the license. The issue is that a third party cannot impose a license on a work simply by putting it out there with a file that claims to be a license from the artist. So this brings in the Free Music Archive: they presumably have some evidence that the rights holder did indeed grant the alleged license, and may be able to provide proof. Your argument may be credible, in the sense that you had a good-faith belief that the item was so licensed, and the website would provide a basis for concluding that that belief is reasonable.
If the work was licensed, then the some rights holder would know that, but not necessarily the current one. Assume the artist made a recording, transferred the rights to Company A, who later sold the rights to Company B who is now coming after you. Artist may have licensed it when it was his, and forgot to tell A. A may have licensed it when they sold the license to B. Artist may have improperly licensed it after he sold the work to A (under the "I wrote it, I have the right to do whatever I want" non-legal theory). A might have improperly licensed the work after selling the right to B (maybe by mistakenly including it in a package deal, i.e. via bookkeeping error, rather than ignorance of the law). Or, they may simply have forgotten.
If this is a DMCA takedown notice, the notice-giver could just be abusing the system. But we don't know how you were contacted, so I'll leave DMCA out of this for now.