While Roy Bittan may have played on Making Movies, he was and is a member of Springsteen’s E Street Band.
Elvis Costello and the Attractions:
Elvis Costello: Guitar, vocals
Steve Nieve: Keys
Bruce Thomas: Bass
Pete Thomas: Drums
Yes, it was definitely a temporary arrangement for Dire Straits. The band had replaced Bittan and added a few other players by the time Brother in Arms was recorded.
Also, in the early '80s, The Cure featured four musicians as described in the OP, with Robert Smith doing double duty on guitar and vocals. They’ve changed members a lot in their 25 years, but managed to stay a quartet until the late '80s.
Not sure of their current lineup besides Smith.
What do you know- another English band that fits the topic! But we have to go back a ways!
The Small Faces were:
Steve Marriott (guitar & vocals)
Kenny Jones (drums)
Ronnie Lane (bass)
Ian McLagen (keyboards)
The Doors
Neither the Doors (no bass player as a regular member) or the Band (five members) fit the criteria.
Some that do:
Bee Bop Deluxe (also British)
Bread (in some lineups)-- David Gates, bass; James Griffin, guitar; Larry Knechtel, keyboards; Mike Botts, drums
Focus – of “Hocus Pocus” fame.
Iron Butterfly – “Inna Gadda Da Vida”
Jo Jo Gunne – great, though obscure U.S. boogie band
The Nice – Keith Emerson, key; Brian Davidson, drums; Lee Jackson, bass; David O’List, guitar
The Young Rascals (later, the Rascals)
Savoy Brown
10cc used the lineup, but three played multiple instruments, so there were several possible lineups.
Uriah Heep
Vanilla Fudge
I’ll add to Reality Chuck’s list – all these are American banmds, to boot:
Til Tuesday with Aimee Mann on bass/lead vocals.
Winger (!)
And suprisingly enough for their genre, country legends Alabama – though their keyboard player plays violin on some songs.
The Doors did not have a bass player, Manduck. Ray Manzarek played the bass parts on his keyboards, although the band also used session bass players on some of the albums.
It’s this guitar, drums, keyboards combo that is really unusual.