No, I wouldn't change it. "Were I he/him" is quite rare, while these are more popular:
7 from bbc.co.uk for "if I were he" 154 from bbc.co.uk for "if I were him" 72 from nytimes.com for "if I were he" 479 from nytimes.com for "if I were him"
This is one of the few reasonable ones with "Were I he" at the New York Times. Quite old too (1915):
FINE POINTS IN AUCTION BRIDGE; Answers to Some Interesting ...
'Were I him ...' is emphatic inversion. It is quite correct if it is used approprietly.
I think, 'If I were he' is American. When you call them, they say 'This is he'.