If I burn BTC using blockchain.com, how can I send the XCP from the blockchain to another address?
You can only burn from using blockchain.info. To send you need to do this with your counterparty client
I'm confused, I thought the address that burned the coins would get the XCP, so if I burn through the blockchain, how can I transfers those coins?
Or would I need to import the private keys for the blockchain address into the counterparty client to send them?
Yes. You will need to import the keys to your local bitcoind/qt client running with counterpartyd.