How about this: A secondary plot in which Clifford arranges musical acts, possibly at the KMUP studios. They're broadcasting radio from there, Clifford is the DJ and the Electric Mayhem would trade places with Solid Foam for the instances when they don't have a musical guest like Green Day. I think that fits in most of the ideas people have had there recently and it'd allow for lots of musical numbers.
I think it's possible for Clifford to be part of the main group of muppets, or at least the second-string group. The way he was used in "Muppets From Space" seemed appropriate for me: Hanging out with either Kermit or Pepe and acting as the occasional go-between for gags, for example.
Clifford: Yo, Kerm, Were you waiting for some painters to show up?
Kermit: Yes, I was.
Clifford: Well, they're leaving, Animal bit one of them.
(I know this isn't a direct quote.)
So probably there'd be moments where he interacts with the Muppets early in the show and then he says, "Well I gotta get going," and head off to do his usual thing. That's Clifford for me.