You’re mixing up “can’t be proved” with “I don’t know how to prove it.”

Every Bitcoin Core node stores when it first saw a tx (getrawmempool true). Mempool.observer, forkmonitor.info, etc. archive those feeds. If my tx hit the network at 17:12:03 and yours appears at 17:12:27, the gap is public and cryptographically tied to the txid.
Deterministic ECDSA RFC 6979 forces each signer to use a different nonce r. Two people who really derived the key produce two distinct signatures; a copy-paste front-runner re-broadcasts the identical sig. Comparing scriptSig bytes is enough to show who computed and who plagiarised.
I can sign an arbitrary message (“Block 850 000, puzzle X, I own this key”) and timestamp that hash with OpenTimestamps or even into the blockchain days before broadcasting the spend. When the puzzle falls, I reveal the signature; the hash already on-chain nails the timeline.
Courts deal with timestamped digital evidence every day e-mails, server logs, CCTV metadata so spare us the “no court on the planet” flourish. In practice the front-runner’s best hope is anonymity, not legal theory, because the maths makes the order of discovery trivially auditable.
Since you're acting in bad faith, I certainly wouldn't want to be your friend IRL, you might steal things from me...

Timestamps of a transaction is not included in the transaction, so the gaps you mention and any logs of any public archives are the time when the TX was seen by that node.
So it's basically not a proof at all. What you call a replacement TX might as well be the first TX that some archival node sees.
And I have no idea what you are talking about regarding the identical signatures. The nonce is random, or it's created deterministically based on the TX data, so unless I'm sending out stuff to the SAME address
(and same value), it's gonna be a different nonce -> different signature.
Actually, I think it would need to be the exact byte-by-byte TX, in order for signatures to be identical.There is no way to actually prove that my TX was not actually created and transmitted before your TX.
The timestamp of a TX is, eventually, the timestamp of the mined block. Any other timestamps are not legal proof of anything, as there's no way to validate the TX broadcasting timestamp.
It's also very funny how you would think that sending some signed message days before the actual TX would serve any legal purpose at all, because, unless you're signing it with the key you're about to spend, it can be basically sent out by anyone. So it's futile as it doesn't proof anything. And if you do sign it correctly, you already know what would happen, since signing it exposes the public key.
I feel like we're going in circles with what "not your key, not your funds" means in the context of a public blockchain.