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That's not a problem with LastPass, it's a problem with password managers - also coincidentally a feature of them.

In case of KeePass at least, you can skip the master password and use keyfiles or HOTP such as Yubikey instead. Or, better yet, you can combine the master password with one of those.

How do you now mitigate this?

You can't.

The DB is encrypted locally using your password (wich lastpass does not have because they store only the hash)

Write it down, store half of it in a safe, the other half in a compartment in your purse/wallet (which you keep on your person).