By cleaning the breech plug, I think that you mean the
inside of it since it should never really need to be removed.
The TC rifles have a patent breech with a powder chamber
that's narrower than the bore diameter. Some think that it's
self cleaning in that not much powder residue accumulates in
it because of efficient burning. It's also because during the
routine "pump and flush" cleaning regimen that most flintlock
shooters use to clean their barrels, most of the residue is
flushed out. But take note that usually isn't the case unless
the breech is soaked for up to 15 minutes or so to first loosen
up the crud. It can be soaked by plugging the vent with a
toothpick or a feather and putting some water in the barrel,
or by putting the barrel breech in a bucket of [warm soapy] water.
The vent liner could be removed to promote more flushing action
to help the debris exit through the vent during the pump and
flush. Sometimes residue will build up directly behind the vent
liner.
Then there are breech plug fouling scrapers which is a ramrod
attachment that a person needs to determine the correct size
to use to fit into their powder chamber. If the powder chamber is
too narrow for a breech plug scraper to enter, then a smaller
diameter bore brush can be attached to a cleaning rod to reach
in to scrape the sides of it.
Using a fouling scraper is not always routine unless there's an
accumulation of powder fouling residue left over from previous
shooting sessions or if shooting a lot, but it's not a bad idea to
use it to check whenever cleaning.
A wide fouling scraper can also be used to clean the face of
the breech plug if any residue is accumulating there.
Here's pictures of several breech plug scrapers and the rounded one
in the middle may be a TC but I don't know if it fits the flintlock.
Below aren't diagrams of the TC breech plug but they show where
the powder residue can accumulate.