Why did The Beatles only record one show tune?

I could never see the difference between a beautiful melody and a cool rock ’n’ roll song. I learned to love all the ballady {sic} stuff through my dad and relatives, Paul McCartney

All the Beatles grew up hearing the show tunes loved by their families. Jim McCartney played them (semi) professionally. He would use standards from the 20s & 30s to demonstrate the harmonies for his sons.

In their early years, the band generally avoided playing songs from musicals.  On one occasion, at The Cavern, Paul sang Over the Rainbow. John stood behind, pulling faces and making faces. The unwritten rule was that show tunes were not cool. They were only played strictly for laughs. Ditto for jazz.

Latin 

They did perform one Mexican song loosely based on an operetta aria. Besame Mucho was enormously popular in the 1950s and became part of their Hamburg set. They even played it at the infamous Decca audition:

But the one Broadway song that would eventually end up part of their official under the wire via Paul’s older cousin, Betty Robbins. 

McCartney only discovered that Till There Was You came from The Music Man after it had been recorded and released on With The Beatles. By that point, there was nothing John could do about it.

The Beatles Don't Do Showtunes? - 4-minute free read