I tried it for my gr-gr-grandfather ― after seeing this discussion ― and removed it again. It looked weird between his names. Instead of Judge [name] [name] (the way it always appeared in print in the newspapers), it shows as [name] "Judge" [name].
It's not a nickname. It's a prefix. But if it's not allowed as prefix, I'd rather leave it off. (And I ain't gonna get into the prefix vs nickname "discussion" again. I'm not recovered from the last time.)
For yours it seems to work, but my "the Judge" was more often referred to as Judge [name] [name]. (I think his son was Dr [initial] [name] or Dr [initial] [initial] [initial] [name].)
(Some ancestors can be a pain! )