Good catches on those, 71dude. It's possible that the use of "We've Got Tonight" might have been for the benefit of us, as the viewers of the show. If it was used as "transitional" music, then that may have been the case. As long as the characters on the show didn't "interact" with the song in any way. In other words, they didn't sing with it, or dance to it, or anything like that.
There was a similar example of this in the opening scene of
American Dreams. It is 1963, yet we hear Stevie Wonder singing "Uptight, Everything's Alright," which didn't appear until 1966. However, since Meg and Roxanne didn't interact with the song in any way, that was, to me, an acceptable use of the song.
However, a couple of seasons later, it was 1965, and the Spencer Davis Group appeared on the revisionist version of
Bandstand to sing "Gimme Some Lovin'," which wouldn't become a hit until 1967. This one especially pains me, because even in 1967, Steve Winwood was still only 19 when it became a hit.
I also previously commented on the fictitious appearance by Rick

Nelson on
Bandstand, since that never actually happened, due to Ozzie Nelson being opposed to it.