I saw CATS twice; once in New York and once with the touring show. It was fascinating to see what they had done with the staging for the touring show--obviously with a show running as long as CATS did, they got to build up a "permanent" set that was completely amazing, and the touring show had to have all manner of clever compromises.
If I am not completely insane, I believe that Mr. Mistoffelees originally sang his own song--yes, I just Wiki'd it and he did.
From Wikipedia--
"Mistoffelees performs the most difficult choreography in the show, including 24 fouettés en tournant and dancers cast in the role frequently have extensive ballet and gymnastic experience. Because the role's technical demands necessarily trump the performer's vocal abilities, the show permits multiple vocal tracks for the character--thereby allowing some dancers to sing quite a bit, while others do no singing at all."
So sometimes the actor who does Mistoffellees sings his own song, and sometimes he doesn't and Rum Tum Tugger sings it. Which means your son was spot-on and right in doing it himself, Misheru.
Here's the wiki-link
http://en.wikipedia.org/wiki/Mr._Mistoffelees