I’m pretty sure Costello considers the music on King Of America to be "REAL Elvis music", and 14 of the album’s 15 songs are Attractions-free.
As I said above, KOA is Elvis’ own favorite of his, and the musicians who play on those 14 songs include Mickey Curry, Jerry Scheff (Elvis Presley, L.A. studios), T-Bone Wolk, Mitchell Froom (lots of album productions), Jim Keltner (John Lennon, Ry Cooder, Bill Frisell), David Hildago (Los Lobos), James Burton (Ricky Nelson, Elvis Presley, Emmylou Harris), Ron Tutt (Elvis Presley), Michael Blair, Jo-El Sonnier, Ray Brown (the great Jazz bassist), Tom Canning, Earl Palmer (the master New Orleans drummer, heard on the recordings of Little Richard and other 1950’s Rock ’n’ Roll greats. Also a member of The Wrecking Crew, and the L.A. studios in general), and the album’s producer T Bone Burnett. Rather than being "mellow", they are amongst the hottest musicians in the world.
And then there are the albums Costello made with The Brodsky Quartet and Burt Bacharach. Do those albums contain "REAL Elvis music", or are they also "mellow"?
Prior to emerging as a solo artist, Costello himself had been in a Pub Rock band, Flip City. Costello and the other members were fans of The Band, The Grateful Dead, and yes, Clover. Producing My Aim Is True was Nick Lowe, a member of another London-based Pub Rock band, Brinsley Schwarz (who later went on to be Graham Parker’s original band).
My problem with The Attractions is two fold:
1- Many UK bands tend to play either slightly behind the deepest center of the "pocket" of the song (dragging), or slightly ahead (rushing). The Attractions are of the latter variety. They tend to rush through each song, which is not the same as playing a song at a brisk tempo. Musicians know what I’m talking about, and I hope the concept is understandable to non-players. Rushed playing is akin to the way some teenage boys make love to a girl 
.
2- The sound of their instruments. While Bruce Thomas’ bass sound fine, Steve Nieve’s organ sounds terrible. He doesn’t use a Leslie rotating speaker with his electronic organ, the speaker ubiquitously seen partnered with Hammond B-3 organs. His organ sounds just like those of the garage Bands of the 1960’s, real "cheezy", and completely lacking in gravitas or soul (very "white" ). And Pete Thomas’ drums sound anemic, with no resonance or depth. Real thin and gutless, lacking tonal color.
Huey Lewis of Clover can be heard blowing ferocious harp on several of Dave Edmunds’ albums, and the Brinsley Schwarz band on two live tracks on Dave’s Subtle As A Flying Mallet album, taped in a pub in Wales. The playing of the BS band on those two songs is red-hot American Rock ’n’ Roll, similar to the playing you hear on King Of America. "Mellow"? Quite the opposite.