Is there any specific reason why you want that method? Goin' with electrum's 136 bits is more than fine.
My reasoning is: if my 12 words are compromised via a physical intrusion, the extension stored in a separate location will guarantee that my wallet is not compromised. Simply splitting the 12 words in half and storing 6 words separately makes a brute force attack easier (I think). That is why I ask.
Each electrum seed is already extended with the word “electrum”, if you choose to extend it more with another seed phrase it'd become “electrum<seed_phrase>”.
If each Electrum seed is already extended with the word "electrum", then why don't we have to input that as a seed extension when recovering a wallet with Electrum or another Electrum compliant wallet?
So it doesn't double the entropy, instead, the entropy remains the same. What it does change is a salt. Once you're done with the seed generation and salt selection, the result is being put through a key derivation function called “PBKDF2”. But, you can of course do it, it'll provide around the same security for a human being.
Are you saying that an attacker would try to brute force the output of the PBKDF2 key derivation function, thus it would be the same difficulty? I would imagine if the attacker were trying to guess words and extension words, then it does double the difficulty.