I looked more on the submitToMainstay flow.
For now, I think we can do something far simpler for the first releases.
Every 24 hours, take all the orderId from the latest 24 hours.
Concatenate them in a single data byte vector sorted by alphanumeric order.
Hash the vector with an hashing function (e.g SHA256 or SipHash).
Commit the 32-byte hash output in an OP_RETURN bitcoin tx output scriptpubkey.
Broadcast and confirm the bitcoin tx.