The studio musicians were all members of Local 47 of The American federation ogf musicians and of the second wave session elite in the Los Angeles now known as The Wrecking Crew and played on hundreds of hits.
I have seen the following members live Steve Douglas (saxophone) James Burton (Guitar) Mack Rebbenack (Kyboards) but this was long after their session careers.
Joe Saraceno was one of a number of producers who would pump out records under a variety of group names and if it started to hit would start to assemble touring bands. With instrumental groups it was easier to tour several bands to promote the record across the states.
From BB info: The Routers - rock and roll instrumental group formed in LA.
Mike Gordon, Al Kait, Bill Moody and Lynn Frazier. Produced by Joe Saraceno.
B. Bumble & The Stingers - LA sessionmen that included Ernie Freeman (Piano), Tommy Tedesco (guitar), Earl Palmer (drums) and Red Callender (bass) on their first two hits. "Nut Rocker" first recorded by Freeman, Palme and René Hall (guitar). Re-recorde version featuring Palmer, Hall and Ali Hazan on piano. Pianist R.C. Gamble led a touring trio from Oklahoma as B. Bumble & The Stingers. A Stateside EP "Piano Styles Of B Bumble" lists Ron Brady (drums), William 'B' Bumble (guitar), Fred Richard (piano).
Looking at the line-ups there doesn't seem to be a link between the two.