Receiving writing credit for a song on an album liner or promotional material does not prove that the credited author actually wrote the song or even that he contributed to its authorship. It is common enough for a star who records a song to be credited as its author even if the piece is actually written by a ghost writer, or purchased from another artist with the explicit agreement that the real author will go along with the deception. In a work for hire the original copyright will go to the non-author as well as the fictitious "credit" or attribution. But work for hire has a couple of wrinkles. If the author is actually an employee, then the writing he does in the normal conduct of his employment never belongs to him. But if he is an independent contractor, in order for it to be a work for hire, it must not only be sold under a contract that says in writing that it is a "work for hire," but courts have also said that the creation of the work must actually begin after the signing of that contract. So if Beyonce wants to buy your song that you have already written, she cannot own the copyright (which you own as soon as it is fixed in tangible form) by just having you sign a contract to do that song as a work for hire after the fact. If she wants to buy the copyright, she needs to acknowledge that it is yours to sell, and have you sign an assignment or conveyance of your copyright to her. She can then register that conveyance and claim to own the copyright as a non-author. To actually claim to be the author when you are not on a copyright registration is fraudulent and may invalidate the copyright at a minimum regardless of whether the true author is willing to go along. It is not inconceivable for a ghost author to have the copyright registered in his name and someone else to claim authorship on the album cover, which after all is not a sworn legal document.
In any event, it is imperative for you to distinguish between whether the band agreement of how the authors would be determined affects the author's rights. If the agreement was just that the band agrees that you will be considered the author for the purpose of giving you credit on the album cover, that does not have much legal force. If the agreement says that you will be considered the author for the purposes of distributing royalties received by the LLC, that affects how payment is divided (a contractual issue), but does not necessarily affect copyright ownership. If the agreement says that you as the author of the song, are permitted to copyright the song in your own person, that may not add much to the rights you already had as the author. The only reason the band would need to agree to permit you to register your copyright individually as the author is if they already owned part of it.
So the question is, did they own part of your song? They could own a part of it either by being co-authors of a joint work, or because somewhere in your band agreement the members agreed that the copyright of anything written by any band member was assigned or conveyed or transferred to the band or its LLC. If the agreement to let you register your song is worded in such a way that is is clear that the other band members were assigning their copyright ownership (if any) in your song to you, then your copyright is undisputed and undivided. If the copyright was ever assigned to the LLC, then lacking a clause that retained some rights to the author, the LLC becomes the owner of the copyright and you need its permission to use it in any way. In that case how would you get your copyright back from the LLC? The LLC would need to re-convey the copyright to you. The exact wording of how they permitted you to register the copyright of your song may be important here, in that it may or may not meet the legal requirements for an assignment. Or it may be that the wording of the band's agreement never assigns the individual author's copyrights to the the LLC, and the agreement to permit the individual author's to register their own copyright is superfluous and has no legal effect.
Now none of this really addresses the issue of the ownership of the arrangement that was recorded on the band's album. The band could have agreed to let you copyright the song composition, meaning the basic melody and lyrics, without necessarily assigning their copyrights of individual parts of an arrangement of your song that was recorded on the album. You have been pretty clear that the guitar part on your song in the band's album was not entirely your sole invention, so the guitarist is probably either the sole author of his own work (possibly a derivative work of your song that he did with your authorization) or is a co-author of your joint work. So did he license your new intended use (could be an oral agreement, and not needed if you are a co-owner), or did he assign his copyright to you (must be in writing, and would prevent him from using the guitar part himself without your permission)? As mentioned previously, Unless the wording of the band's agreements is so crystal clear as to be beyond question in interpretation, or if it is ambiguous that the people who signed the agreements will testify that their understanding of areas of ambiguity is the same, then you are going into the tall grass here.
If you are sure that your author's copyright in the original song has never been conveyed to any other entity in an assignment, or if it has that it has been re-assigned back to you, then you can do a new arrangement of your song without asking anyone. In that case, if you are sure there is no surviving obligation to share revenue from your song with the band or the LLC lurking in something you signed in the past, your new arrangement can be done without paying anyone else. If you use the arrangement or portions thereof that you are not the sole author of, then you may be obligated to get either the other author's permission or pay other authors or entities or both.