If I may speak from a purist's point of view ...
... you aren't going to become Red Samurai.
Red Samurai are the super-snootiest of the super-snooty-Japanese-corporate-bodyguard, created specifically by the CEO and kept by him as a pet project for the longest time. Considering the origins of your background - born and raised in Seattle, minor Yakuza gang thug (not even a full-bore Yak yourself), it really isn't the sort that would be allowed to even try out for the Reds, much less be accepted.
The Red Samurai are some of the most loyal corporate security that exists - on par with the Jaguar Guards for Aztechnology, or as the Seraphim were for Cross Applied Technology. (Sorry; I can't think of any others with that level of loyalty.) They were born in Renraku of loyal Renraku citizens, were raised in Renraku day-care, went to Renraku gradeschool, graduated from Renraku high school, dated a Renraku girl, think metahumans are stinky-poo-poo (though they're allowed to actually look in the direction of elves without pulling the trigger these days), went to Renraku security, married a Renraku sweetheart, had Renraku children, and proved that they bleed Renraku red and preface every thought with 'Renraku!'
Get it?
I'm a little iffy on the whole 'Yakuza gang' thing, and 'being bought out'. A member of the Yakuza, sure. And I can imagine that he and a double handful of age-mates stuck together, got their tats together, went through the ceremonies together, but during one or another of the wars - the Mob War in 2058, perhaps, as you suggested - one by one the rest of his 'gang' got killed in action against the Mafia, Triads, or other syndicates or support groups, with Fumetsu getting 'cheaper' cyber replacements (as compared to fleshy ones) in order to recover from injury each time. This 'resistance to death' would earn him the nickname, I'm sure. And afterwards, as the last man standing ...
... does he have to be wanted by the Yakuza? Shotozumi wanted to break free of Tokyo, to create his own rengo; perhaps you were a Watada loyalist, and chose to instead go 'back' to Japan. That move could have gotten you in place for improvements and trade-ups on your implanted cyberware as your slightly-American way of looking at things allowed you to get the job done just a bit better than the next guy; that sort of thing gets you noticed. And with that, you got harder assignments, especially as your talents as a kick-artist, wetwork specialist, hit man, assassin (a rose by another name...) got you recognized. This put you in place to be a major hitter during the two-year war (2066-68) between the Watada-gumi (home sweet home) and the Mita-gumi, one which (apparently) eventually broke the Mita-gumi's back and kept them as part of the Watada-rengo. (Reading between the lines in Vice, it sounds like the Oyabun of the Mita-gumi was assassinated, leaving as his heir a child - which event would 'calm things down' considerably.)
So why are you then out of Japan? Maybe you did something that put you into disfavor. Maybe you expressed an opinion that, while not so dishonorable as to require a 'token of apology', meant that you were better off 'sent away for a time'. Serving the Shotozumi-rengo in Seattle would be somewhat dishonorable too, or at least leave a bad taste in your mouth (considering they might, in your mind, still be traitors of a sort), so you're currently on your own, with only a contact or two who preferred the Watadas to the Shotozumis.
And now, to make ends meet and to fill the time until you get called back to Japan, you take the 'odd job or two', i.e. shadowrun for a living. And really, even if you are hired to target one of the local Yak operations, it isn't as though they don't deserve it for breaking away back in '58 ...