I hope I am wrong but the sig doesnt look right to me. Also Tolkein's sig was beautiful , rather like the elvish scripts and he usually used a fountain pen
I've only seen Tolkien signatures in nib pen, like a fountain pen, and those are typically signed slower and more carefully than a ballpoint and can have some different characteristics. That said, there are characteristics in yours that I haven't seen in known genuine signatures. For example, the "RR" typically slants right, but left on yours. That's something that's usually consistent in signers no matter what writing instrument is used.
I'd say talk to an auction house who deals in them or a known Tolkien specialist and ask what they think.