NO! I think you misunderstand what "template matching" means here. This isn't jamming some other signature format into a BIP 322 signature. There's no "template friendly message" to create, and no signing again.

In this context, template matching occurs against the output script for which the BIP 322 signature signs for. This output script is produced by the address provided, as addresses encode output scripts. A template matching verifier would simply look at the address, see if that address type is one that it understands. Because standard scripts have fixed witness structures, the verifier can look at the scriptSig and witness to see if those match the template, then extract the components that it expects, and verifies that those are correct. This also generalizes to things that don't have addresses, such as Lightning HTLCs, as the script has a fixed structure and so the witnesses will also have a fixed structure.

For example, a template verifier for P2PKH would be given a P2PKH address. It understands P2PKH addresses and knows that it should expect a scriptSig that begins with a byte between 0x07 and 0x49, followed by however many bytes that number was, followed by a byte that is either 0x21 or 0x41, followed by however many bytes that number is. It knows that the first of those is an ECDSA signature, and the second is a public key. It knows that it should check that the public key's hash160 should match the one encoded in the provided address. It knows that the signature, pubkey, and the sighash of the current transaction, should validate using ECDSA. It can do all of this, because the programmer hard coded these things in. If anything doesn't match, it just immediately says the signature is invalid. If it gets an address it doesn't know, it tells the user that verification is inconclusive because it doesn't know that address type.