To get the private key from your wallet.dat without fully sync you need Bitcoin core installed on your PC and make sure the prune mode is disabled you can check it on the bitcoin core settings to disable it and close it and open it again.
I don't understand why you should need to disable prune mode for exporting your private keys.
Whether punning is enable or not, it has nothing to do with the keys stored in the wallet and you should be able to export the private keys.