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Tournament rating is probably an interesting thing, and it might be worth maintaining it for use in draws, and just to have it.
Using any kind of rating as an additional criterion when indicators are equal is incorrect, if only because it puts tournament participants in unequal conditions from the start.
I wonder, is rating used in this way in any sports? I can't recall any such examples. Except perhaps weighing in weightlifting, but there it's the opposite - the lighter one wins.
I suggest using a simple, quick, absolutely fair (i.e., giving all participants equal opportunities) and limiting the possibility of match-fixing method - a coin toss.
In football, for example, a draw is used in some situations when indicators are equal.