You write as if this exchange wouldn't own the same address on the other chain then too. But they would ideally. Well, probably depends on the way their wallet creates addresses. Core might be not good if i remember correctly.
No - you can't do this - the person who is sending you (being the exchange) has the private key for those UTXOs (not you the exchange).
So it is within their potential ability to "double-spend" (not you the exchange).
Clear yet?
(why do I get the feeling you don't even know what UTXOs are)