I realize San Clemente is a ways from Fullerton at opposite sides of the county, but "House of Bibles" on Commonwealth is where I would check. (They're closed right now, it's late, or I would have called to ask about what they'd recommend about repairs.)
I lived in Fullerton for years (still live close enough to go) and spent a lot of time there, the owner is very knowledgeable, and I'm sure could answer questions you have about the repair needed, advise if you should send it out of state, etc., so why not give them a call in the morning? Hopefully you can get some additional information.
Personally, I would be hesitant to send an heirloom Bible anywhere, I'd have to know where it was.
House of Bibles
123 East Commonwealth Avenue
Fullerton, CA 92832-1907
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Hours: Sun Closed; Mon-Fri 11am–7pm; Sat 10am–6pm