If black needs a draw in the last round to win the tournament, which
opening should black pick?
If a player only needs a draw in the last round, he or she should play normally (of course this is easier said than done) and only sometimes remember that to reach the goal they only need a draw, for example if they can force draw by perpetual. This means the player should play the opening they are most comfortable with. If the only opening black knows is Sicilian Najdorf, then they should play that.
Moreover, I don't think there is really a drawing weapon for under 2000 players; I believe draws a fairly rare at that level. Having said that, I think that the Berlin is more "drawish" in the since that it leads to exchange of queens (a least in the main-line). I don't think that the Petrov is really all that drawish, usually the position is pretty open with a lot of piece activity and opportunities for tactics.
The reason the Petrov has a drawing reputation is that many top-level GMs use it to draw. However, the main reason it works for them is that they know the Petrov really well, and often can show majic with tactical nuances, not because the position is really drawish or super solid. Even Kramnik had some quick losses with the Petrov (see Anand-Kramnik below), when the opponents had out-prepared him, those positions are tricky.
The Marshall gambit in Ruy Lopez is another opening top-level GMs often use to draw; however, that opening is hardly drawish or solid for sub 2000 players.
Here is is Anand-Kramnik game mentioned above:
[Event "Mtel Masters"]
[Site "Sofia BUL"]
[White "Viswanathan Anand"]
[Black "Vladimir Kramnik"]
1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nf3 Nxe4 5. d4 d5 6. Bd3 Nc6
7. O-O Be7 8. c4 Nb4 9. Be2 O-O 10. Nc3 Bf5 11. a3 Nxc3
12. bxc3 Nc6 13. Re1 Re8 14. cxd5 Qxd5 15. Bf4 Rac8 16. Qc1
Na5 17. c4 Qe4 18. Bd1 Qd3 19. Re3 Qxc4 20. Re5 1-0