Goodness. Neither interpretation fails. Entangled states look like ordinary probabilities from inside, so you can never be both at once. Correlation with the past preserves memories in both interpretations, so you'll never remember being in a different "branch" (MWI) or a different column of the matrix (The Matrix! I mean typical linear algebra interpretation).

In fact, now that I think about it, the measurement collapse is probably equivalent to starting with the continuous, deterministic big ol' wavefunction of the universe, and then asking "what does it look like to an observer from inside?"

Copenhagen fails. W.r.t. Wigner's friend, isn't that entire thought experiment highly flawed when one considers timeless physics?