When cleaning old cake I'll use a couple of methods, but in the case of a noticeably uncentered cake I'll wrap glass paper around a piece of doweling, or a Pilot marker pen and work to carefully remove the cake until it's just a thin veil over the wood.
Hopefully the chamber walls will be solid, but don't be surprised if cracks show up. Those can be dealt with later. Just get the cake off and scrub out the airway with careful attention to the mortise. Dental picks can be useful for getting hardened crud out of the corners. Every bit of crud needs to go. A nylon shank brush will replace a number of pipe cleaners for the process of cleaning out the airway.
Give the pipe a S/A treatment to remove further sediment, tars, and oil.
If there's a lot of lava on the rim, softening it will definitely help with removal. To get rid of lighter amounts of lava, spit and a rag, or cold coffee and a rag, coupled with elbow grease, should do the job. With a blasted rim, a soft toothbrush can be useful for getting into crevices.