No the transaction is within the bitcoin core wallet that you have running on the node. If you have the actual TX that you can see on a block explorer it's out of lightning (2nd layer) and into the 1st layer itself.
So all you have to do is spend that transaction with the CPFP.
How do you have this setup? You should have some access to that wallet to send the funds.
-Dave
I might be wrong, but I don't think CLN uses a bitcoin core wallet? bitcoin-cli listwallets returns an empty array for instance. Also CLN does not seem to be aware at all about what is going on in the mempool.
I thought it did, could be wrong I have never tried with CLN but it was in the back of my head that unlike LND it did.
For LND people have written tools to get to the keys
https://github.com/lightninglabs/chantoolswalletinfo Shows info about an lnd wallet.db file and optionally extracts the BIP32 HD root key
Not sure if anything similar exists for CLN
Sorry I can't help more.
-Dave