I think you're on the right track regarding Monk Montgomery and the Fender Precision Bass. Since the latter was introduced in 1951, and Monk was with the Lionel Hampton Big Band from 1951 to 1953, it's possible that the first jazz album with electric bass was one of Lionel Hampton's.
This site seems to have a better LH discography than Wikipedia:
Link Removed
including six released in 1953. However, since it doesn't list 1955's Hampton-Getz album, it does not appear to be complete.
Nevertheless, by cross-referencing to Amazon and other sites you may be able to eliminate the albums on which others play bass, and focus on those remaining.
Add-on:
Continuum Encyclopedia of Popular Music of the World: Performance and Production, by John Shepherd, which is (at least partially) searchable via Google Books, states that the first electric bass recording was by Monk Montgomery with Hampton, in 1953. But that electric bass was used, albeit unrecorded, by Vernon Alley in 1940. Obviously Vernon did not play a Precision. Perhaps it was a Tutmarc.
If any recordings were made using a Tutmarc bass, perhaps Paul Tutmarc's grandson Greg would know. Greg's e-mail address is available here:
http://www.talkbass.com/forum/showthread.php?t=109056
Here's a brief Tutmarc thread:
http://www.talkbass.com/forum/showthread.php?t=671561&highlight=tutmarc
And an oldest bass recording thread:
http://www.talkbass.com/forum/showthread.php?t=109056