If I use Electrum to generate a 12 word seed, and then generate a new wallet with a different seed, can I use the first 12 words as the seed extension for the new wallet?
Yes.
Would this effectively double my entropy?
No.
Can I store the 12 word seed and the 12 word seed extension in two different places safely?
This is the only way you should store them. Storing both your seed phrase and your seed extension together renders the seed extension nearly pointless, since if an attacker compromises your back up they immediately have both and can take your coins.
Are there any major flaws in this method?
Not really. Using a seed extension is a good idea, and by using a randomly generated seed phrase as the extension you can be sure that it is complex enough to be resistant to brute forcing. The only issues would be human error - getting confused as to which is which, making a mistake when writing them down, etc.
Would using a multisig wallet be better?
That depends on what you are trying to achieve. A seed phrase with an extension provides protection against one of those two back ups being compromised, but doesn't protect against your wallet itself being compromised. It does however keep your transactions small, and can also give you plausible deniability (depending on how you use it). A 2-of-3 (for example) multi-sig protects against one of your back ups being compromised, and protects against one of your wallets being compromised, but requires more complex back ups and results in larger transaction sizes (although not for long once Taproot is activated).