Berlin Defence or Petroff? Which one is the better drawing weapon? 

Which one is harder to beat? 

Which one is more solid?

Which opening should black pick if he/she doesn't want to lose?

If black needs a draw in the last round to win the tournament, which opening should black pick?
There no certain answer to your question. It depends.

An opening with a "drawish reputation" usually turns against the player who chooses it without having a deep knowledge of the opening itself.

That said, I think that both of these, if understood, can be a very hard match against a White player who desperately tries to win. But they're not easy to play.

In my opinion, as a 1. e4 player (so, with the "opponent's point of view"), the Berlin is quite solid but harder to play: Black has to find suitable squares for his pieces, and must play carefully to avoid White attacks.

As White, I am comfortable playing against the Petroff: for Black it's a little bit easier to play, but has many traps, and usually the Knight becomes an easy target for White pieces. Black will lose tempo while trying to defend it.

In conclusion, don't choose an opening only for its "drawish" reputation: this may be true only at higher levels, where players know very well almost every opening. At lower levels, on the contrary, the player who relies on a specific "drawish opening" in order to draw a game will find himself in trouble more often than not.