I don't really follow the Bitcoin Puzzle scene, but assuming the public keys for these particular puzzles are indeed within a range that could be derived within seconds, digaran already answered the question following the post you were linking:
you'd need to either have a bot that automatically double spends the tx or talk with a large mining pool to privately include your tx in a block
Though even that will not guarantee success if your adversary is able to broadcast their transaction at the roughly the same time (i.e. within seconds) as you: (1) They may attempt multiple double spends so now it just becomes a competition of who spams the network most effectively and (2) the mining pool you are colluding with may not be the one actually find the next block. Nonetheless following either approach would increase your chances over just hitting send and hoping for the best (though as mentioned above I can not asses the risk of getting looted).