Any (to a normal mortal) fatal blow to Connor will disable him long enough to deliver the (permanently) fatal blow.
I don't think that's true.
In the film, we witness a rapier duel, where Connor is run through the heart several times, (certainly a fatal blow to any other mortal), while Connor just stands there, drunk out of his skull, and laughing manically at his bewildered opponent. Eventually Connor gets bored, staggers off, and mutters his apologies to the frustrated duelist...
We watch him cut his hand open, without showing signs of significant pain, or going into any form of shock, and we watch it heal completely in seconds, which is admittedly a long time in a duel. If Musashi cut him in half, Connor would probably just blink, and cut Musashi in half in return before Musashi could pull his sword out.
And it's not even guaranteed that Musashi is a better swordsman: Musashi only knows how to defeat the fighters of his day. His "new philosophy" of fighting might be new to the people of his era: but surely not to an immortal who studied swordplay for centuries, exploring every nuance for advantage.
Pain resistance and kill difficulty aside, Connor has explored every sword technique known to history. Musashi only knows the ones from feudal Japan that he personally invented; and his advantages derived largely from his unorthodox style of fighting; his success was as much a result of superior psychology as physical prowess, though of course he was a master of both.
Connor has trained with a view to decapitate; but that doesn't have to be all he did with his centuries of training. He's trained for centuries to find advantage, just like Musashi: expecting to be attacked at any moment for centuries gives one a professional interest in swordplay. Musashi only fought for about fourty years, and his psychological tricks and "unorthodox style" aren't new to Connor...