Can you post a scan?
Sorry about that, I have attached a photo below
Looks legit. If it were printed in the book the company would have most likely used his full "Jimmy Carter" signature. His signed books are extremely plentiful so there's usually no worries if you find one cheap.
Is there any particular reason why he'd generally just sign "J Carter?"
That is typically how President Carter signs books...just “J Carter”. Getting a full name autograph is much more rare.