unless you're running an aftermarket breech plug, your T/C has a "Nock" style breech, as shown in the photos (which are, i'm pretty sure, a flint breechplug)
I have several flintlocks without this system, and they work just fine, but for a while the 'patent breech' was all the rage and any one who didn't have one was a compleat fool, would never have huis gun work properly, would be unable to hit the floor if he fell on it, would be ugly, have his mother dress him funny, and would be roundly mocked by the black powder cognoscenti world wide. Not really sure why T/C went to the trouble to make their rifles with this system, since a plain old flat face breech will work as well (in my experience) ... but it does look "kewl" in the marketing literature, and thos with very little experience in such matters will be impressed ... i know i was ...
as you can see, the chamber gets much thinner after the barrel meets the breechplug. if you want to get this super clean, get a .22 cal brush which will fit down the smaller portion of the system. (i use a synthetic bristle brush - don't tell the P,C, police
)
if you take the barrel out of the rifle (easily done) and put it in a small bucket of warm soapy water, then run a patch up and down the barrel, the water will be pulled up the barrel when you withdraw the cleaning rod, and you can pump water back and forth, thus completely cleaning out you bore. A few dry patches, some good oil, and you're all set. You can even do this at the range if you're so inclined.
Good luck with your new rifle: it will serve you well for several generations if properly maintained. A shame that T/C is out of business.