Designed me a theatre (5)
I believe the answer is:
meant
'designed' is the definition.
(thesaurus)
'me a theatre' is the wordplay.
'theatre' becomes 'nt' (National Theatre).
'me'+'a'+'nt'='MEANT'
(Other definitions for meant that I've seen before include "proposed" , "Deliberate" , "represented" , "Intended, had in mind" , "Aimed" .)