Kate Bush was coming off Hounds of Love in the mid eighties cementing her place as both a great musician and a popular (in the UK) musician. A status she still holds today.
Big Country haven’t lasted quite so well in the memory (in the UK) but they were considered good if not great musicians then and despite not topping the charts were very popular. Albeit they were automatically described as ‘the band with guitars that sound like bagpipes.’
Big Country front man Stuart Adamson (died far too young from alcohol abuse) was already famous for his previous band The Skids who had several hits including the absolutely fantastic Into The Valley. His time in that group led to him being considered one of the best British guitarists to emerge from the punk / New Wave era. You have to consider whether his status would have increased if he had lived longer. Although he died much older than Jimi Hendrix, Janis Joplin and Kurt Cobain.
TLDR: At the time Kate Bush and Big Country were similar status.
AFAIK Neko Case is considered a full member of The New Pornographers, so that’d be a separate category: where a member of a group/band is more famous as a solo artist.
I’m not sure who was bigger at the time(s), but Carole King sang backup on James Taylor’s version of Carole’s song You’ve Got a Friend, and vice-versa. And Joni Mitchell was on both as well.
King and Taylor were both fairly early in their solo recording careers: “You’ve Got a Friend” appeared on King’s second album, and Taylor’s third.
Taylor was, maybe, a bit bigger of a “star” at that moment, as his second album, Sweet Baby James, was a top 10 hit, and had “Fire and Rain,” which was also a hit. King’s first album had not done as well, but she had been a well-known songwriter for over a decade, and had had a couple of minor hits as a singer in the '60s.
So, overall, they were probably at not-dissimilar levels of fame.
(Also, their two versions of that song were recorded more-or-less at the same time, and in addition to having Mitchell on both versions, they also had many of the same musicians.)
On the recording of the song “Basketball Jones” by Cheech and Chong, the singers and musicians include George Harrison, Billy Preston, Carole King, Darlene Love, Michelle Phillips, and Ronny Spector:
Michael McDonald sang backup and co-wrote Kenny Loggins’ “This Is It.” While McDonald was with a more popular band, I think that Loggins had more name recognition.