Is there a time limit how long a signed transaction is valid? When splitting coins from a hard fork and signing the transaction on the forked coin, is there a time/block limit when the transaction gets invalid on the BTC chain, so a potentially possible replay attack gets invalid forever?

Imagine this scenario: You successfully split 1 BTC and 1 BXX (which has no replay protection) from address A to B and C. At some later point you forget about it and send 2 BTC to address A. Can the old 1 BXX transaction be used to replay A -> C in BTC?
Bitcoin is not an account based currency like Ethereum, which track the balance in addresses. Instead it is a UTXO based currency. In a UTXO-based currency, funds are tracked in portions called "Unspent Transaction Outputs", these are uniquely identified through their history, specifically through the transaction id that created them. Any new money moved to the same address would have a different history and thus would not be replayable.