Interesting battle this one.
I think there are a handful of innings in test history that stand apart as the perfect example of a particular type of great innings. You've got the match-saving rearguard like Hanif's triple century or Border's twin tons, carrying the team to a fourth innings total like Lara, turning a game with an aggressive innings like Botham or Laxman, or surviving against all odds to make runs when nobody else can like Gooch or Waugh's 63.
Bradman's 334 IMO is the definitive example of utter domination of a bowling attack. The greatest batsman in history at the peak of his powers, simply dominating a quite capable bowling attack for a whole day like nobody else ever has or ever will. Three hundred runs on the first day of a test is simply amazing, and for all the great Gilly or IVA or Sobers knocks you could name where they tore apart the bowling, none of them ever did it to quite that degree. By all accounts it was a virtually chanceless innings as well. Bradman's finest, along with the 270 on a horror wicket that was knocked out in the first battle by Laxman.
Bannerman's innings is obviously legendary in its own right, but holding the oldest record in cricket only takes you so far. A great knock, but it's difficult to judge the attack or the rest of the context in which the innings was played, and in terms of being the sole representative of an entire side with the bat, there's perhaps better innings, though not many.
Bradman's is more unique in my mind, so it's Bradman for me.