I used to have the vinyl recording of the original soundtrack for this film and it clearly credited Ms. Crain as doing her own singing for " I wanna be Loved by You" with Jane Russell, and believe this might be the only time she did so in one of her musical films. This should be corrected on the IMDB soundtrack listing for this film.
I have the record. The album's cover is "JANE RUSSELL JEANNE CRAIN in GENTLEMEN MARRY BRUNETTES" and the back cover gives no indication of who sings what. On the bottom of the front cover in small print it states "Alan Young, Scott Brady, and Rudy Vallee" and in even smaller print "and the voices of Johnny Desmond, Anita Ellis, Robert Farnon." On the record's label it reads in small print "the voices of Anita Ellis and Robert Farnum singing for Jeanne Crain and Scott Brady" and for I WANNA BE LOVED BY YOU (as you noted) it is marked "Sung by Miss Crain herself." I would imagine the movie used Jeanne Crain's vocal on this song as well, sense it would not make since for the record to have her sing just this one song and Anita to sing the other tracks "for" her if that does not parallel how it was done in the film.