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Is it possible to derive a new 12 word seed from a 12 word seed with passphrase (13th word) such that they both produce the same master key?

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No, this is not possible.

The mnemonic and passphrase are put through PBKDF2. Any change in the input results in a completely different output.

While it is possible that there exists some 12 word mnemonic that just happens to hash to the same seed as your original mnemonic + passphrase, this would require bruteforcing all possible mnemonics to discover, and would not be derivable from the original mnemonic in any form - a hash collision would be completely unrelated.

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