A_hash = Hash(Block Head + sign(Block Head))
You don't have to add signatures to the block header. You can add them to the coinbase transaction, like it is done in signet, in this way you can make it backward-compatible. You can find all technical details in BIP-325:
https://github.com/bitcoin/bips/blob/master/bip-0325.mediawiki.
In general, to start your network, all you need is constructing signet challenge to allow signing blocks only by those miners, which passed the initial selection. By default, signet challenge is set to 1-of-2 multisig, so only two miners can produce blocks (because it is needed to test things in more stable way than in testnet). You can change that signet challenge, just to allow only signatures that passed the first round and you will get your coin.
Thank you very much Garlonicon for the link. Yea. You are absolutely right on A_hash. It was my first intuition and then I switched to B_hash to avoid congesting the network as every miner can do all the B_hash calculation by itself. Your remark on backward-compatibility is brilliant.