In the single elimination portion of a tournament, what happens if a player would lose to the “sudden death” rule while controlling Exquisite Archangel?

My understanding is that they would still lose the game, because the sudden death rule is akin to a Game Loss in that it bypasses all the rules from the CR (in the same way that a Platinum Angel cannot protect you from sudden death), but I'd like confirmation.

Your understanding is correct. As with its predecessors, Exquisite Archangel cannot prevent a player from losing to sudden death rules in a timed single-elimination match.