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>”. 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.
Would this effectively double my entropy?
No.
I might be a bit dense today, deleted my previous post after realizing something.
If I'm not wrong, the seed isn't extended by adding 'Electrum' to it. The salt is however, 'Electrum + passphrase' instead of 'mnemonic + passphrase'. If the seed can be used in the salt to produce a different 512bit output, wouldn't there still be a considerable increase in entropy as long as the ENT of the input < length of the output? I'm sure I'm missing something here.
My main motivation behind this post is to have my backup in two pieces to protect against a physical intrusion. A 24 word seed with higher entropy has no benefit over a 12 word seed if the physical backup is stolen. If I break the 24 word seed with 256 bits into two 12 word parts, can I safely store them in two separate locations like I can with the seed extension?
Yes. In fact, just create a 24 words BIP39 seed for this purpose. I'll consider Shamir secret sharing for some redundancy as well and split them up further.