Great question a real catch 22. Like has been said, win the cup and that team become legends , written into the history books. Get relegated and it can be a long long way back.
Wigan a couple of seasons ago look at them now.
Birmingham City won the league cup, first trophy in their history and were relegated the same season. They played and lost 3-0 to West Brom the weekend after winning the cup ( probably still hungover) Thise 3 points meant West Brom stayed up instead of Birmingham.
The list goes on...
Think I'd still rather the clubs name on the trophy though.