Put more simply, is it possible to set up a multi-signature wallet that takes both signatures from itself?
No, two wallets are not required. It sounds like you might be confusing wallets with addresses, where one bitcoin wallet can house many bitcoin addresses and keys.
If you want to be able to execute a 2 of 3 multisig from one wallet you can do so with bitcoin core with raw transactions. You will have to keep track of the private keys and redeemscript yourself and save them somewhere like a thumb drive for example.
Here's a video tutorial that can guide you through creating and spending with your multisig address using bitcoin core.
You can implement Shamir's secret sharing, in this case you are not technically implementing multisignature. This methodology can be used to divide the private key S of one address into n pieces. Using this method m of n sections of the secret can be used to derive the secret key. This may be used as a comparable solution to multisig when only 1 address is used (this method obviously only requires one wallet)
No. Technically, you don't need two separated wallets.
But as far as i know there is no wallet app that supports your case. Maybe because your use-case doesn't sounds after a reasonable usage of multisig.
The main purpose of using multisig is to get multiple signatures from a) multiple devices (=multiple keystores/wallets) or b) multiple identities (=every identity keeps it's own wallet).