I've recovered an old wallet, where I previously Monero to an exchange from. How can I use the
SpendProofV1, recovered from the
History tab of the GUI, to prove I deposited Monero into their exchange?
The SpendProofV1 string contains a second, newly created ring signature that proves exactly the same input ownership again, but using different random initialization data. Only someone that owned the inputs of the transaction would be able to create this second valid ring signature. That is all the SpendProofV1 string contains. There is no key derivation communicated, because it is unknown by the wallet in this scenario.
Therefore you need to send both the transaction ID and the SpendProofV1 string. The exchange will be able to use this information to verify that you sent the funds.