It seems that the Annesley case exactly fits your criteria.
I've never done this before, but I decided to submit your question to the AI system at Grok. It was interesting to watch it work. It reported that it was considering the Mowbray/Stourton claim to the earldom of Norfolk, the clashing claims to the Earldom of Devon by Viscount Powderham and his cousin, and a few others. It did not find the Annesley case I cited, however.
Grok's final answer was that the closest it could come was the Mar Earldom case. I know it does not fit your criteria, but some may be interested in Grok's response. I include it only to show what Grok came up with in about three minutesm and not because I think it answers your question.
"Yes, there are historical examples in British peerage law where claimants successfully proved their right to one title but failed to prove their right to another considered higher or senior in the dispute.
One notable case is the Mar Peerage Case (1875). Following the death of the 9th Earl of Mar in 1866, John Francis Goodeve-Erskine (the heir-general) and Walter Erskine, Earl of Kellie (the heir-male), both claimed the Earldom of Mar. The House of Lords Committee for Privileges ruled in 1875 that the earldom created in 1565 by Mary, Queen of Scots, was a new creation passing only to heirs-male, awarding it to the Earl of Kellie. Goodeve-Erskine failed to prove his right to this 1565 earldom. However, the Earldom of Mar Restitution Act 1885 declared the existence of two separate earldoms: the 1565 creation for Kellie (and heirs-male), and the ancient earldom (dating back to around 1404 or earlier) for Goodeve-Erskine (and heirs-general). In this context, Goodeve-Erskine proved his right to the ancient title but not the 1565 creation.
Although both titles are earldoms of the same rank, the 1565 creation was treated as the principal or "higher" one in the initial dispute (as it was the one actively used and contested), while the ancient one was restored separately via parliamentary act. Precedence in peerage is typically based on creation date, giving the ancient title higher status, but the case illustrates a partial success in proving rights to titles within the same lineage."