At home terrible strain with fellow being stubborn (12)
I believe the answer is:
intransigent
'stubborn' is the definition.
(thesaurus)
'at home terrible strain with fellow' is the wordplay.
'at home' becomes 'in' ('in' can be similar in meaning to 'at home').
'terrible' is an anagram indicator.
'with' is a charade indicator (letters next to each other).
'fellow' becomes 'gent' ('gent' can be a synonym of 'fellow'**).
'strain' with letters rearranged gives 'transi'.
'in'+'transi'+'gent'='INTRANSIGENT'
'being' acts as a link.
(Other definitions for intransigent that I've seen before include "Unlikely to budge" , "Obstinate" , "Irreconcilable" , "Diehard" , "unwilling to compromise" .)