Your Beatles example is a good one, and lacking some agreement to the contrary, if Paul were to re-record his song using Ringo and John's original compositions, he would need a license to do so. A non-exclusive license could be as informal as George saying, "hey man if you ever want to record that guitar part feel free." If your guitarist said something like that, even if there is nothing in writing and no witness to the utterance, you have all the license you need. If he did say that, and you think he will lie about it or forget he said it, then it is best to get something in writing just because it might be difficult to prove. If something in your LLC papers or other agreements says that you have such an unrestricted license, that would survive the end of the band collaboration, then you are good to go.
The only way you can challenge his copyright to his composition which was clearly designed entirely to complement and presumably derived from your original song, would be to claim that it lacked sufficient originality to derserve its own copyright. If I was given a license by Tom Clancy to write a derviative work of Red October, and I just copied the original word for word and only changed a comma in chapter three to a semicolon, he or anyone else could claim that I did not have a vailid copyright in my own name. If your guitarist just did a simple strum of chords to follow your melody, then that might be an option. But if your guitarist's part that he invented and recorded was so unoriginal, then there would be no point in you copying it in your re-recording. You could just strum your own chords. The fact that it adds enough to the song for you to recognize it as uniquely valuable makes it hard to claim it is unoriginal.
The issue is not who is the true author of the song, but who is the inventor of the work derived from the song. This is not some radical legal weirdness. A whole profession is built around the fact that an arranger can take another author's song and enforce his own copyright of the arrangement he invents to accompany it in order to prevent other arrangers from copying his work, and musical arrangements are specifically named as examples of derivative works in the copyright law. The original author can enforce his copyright to prevent the arranger from ever copyrighting the arrangement, or he can license the arranger to do so, but that license to use the songwriter's work does not give the songwriter ownership of the arrangement. If I loan or rent my skillsaw to someone to build his house, that does not mean that I own his house any more than he owns my saw. The fact that you willingly recorded your song with the band, including his guitar part, is convincing evidence that you knew and tacitly agreed to his creating the part. You have in effect authorized the creation of a derivative work.
http://www.finnegan.com/resources/articles/articlesdetail.aspx?news=9cbb473b-f87b-47eb-8d4b-0202ad56343a If you believe that your contribution to the creation of the guitar part was so specific and material that you can be considered an author of that part as well as the guitarist, and if you and the guitarist were in agreement that it was intended by both of you that, considering just the guitar part, it was "a work prepared by two or more authors with the intention that their contributions be merged into inseparable or interdependent parts of a unitary whole," then you can argue that the guitar part is copyrighted as a joint work. In that case as a joint owner of the copyright you have all of the rights to exploit it that you would have as sole author, and so does he. In that case you do not need his license. Note that if the guitarist and you were not in agreement, then the issue is not clear. You should hope that other band members do not believe that their contributions to your work as a whole makes them co-authors of your work as well. In any case a part owner of the joint work copyright has all of the rights as a single owner to license and perform the work, but is required to share the proceeds of any monetization with his co-author. Unless the formula for the division of the revenue from the joint work is otherwise specified, the law requires it to be shared equally among the authors. The proof against an unintended creation of a joint authorship/copyright is to have your collaborators execute a written assignment of their rights in a work that you claim. That way even if a court eventually decides that they do have a co-author relationship, and hence a joint copyright with you, it will be clear that they have vacated their claim and leave you alone able to exercise the rights in that work.
There are things that are just and things that are fair and then there are things that are legal. But ask yourself, if this thing that the guitarist created, at least in part, is something that you value enough to use in your new recording, is it not fair and just that he share in the value he helped to create?