Why are you listening to a man you know is untrustworthy?
There is absolutely zero you "need" to do about this or the Japanese government would have contacted you.
Yes, that will affect inheritance even if not directly. To get inheritance you need to prove who you are, and the koseki, as far as I know, would be a large part of that proof. One thing Japan is really bad at is options and alternatives. They always, ALWAYS go for the magic bullet and the one-size-fits-all-shoe. They will trust the koseki sooner than a DNA test, assuming they would even discuss a DNA test. Its whacky.
Also, as far as I know, the koseki has ZERO to do with citizenship. I am married to a Japanese and on her koseki, but I am not even a permanent resident.
Basically what I would do is tell your uncle to cough up some official documentation proving his words. After all, is he a lawyer? Just tell him he is incorrect and if he thinks different, he needs to prove it. Its his contention, its his homework, not yours.
Good luck with the inheritance in any case. My experience is that laws and rules aside, those not there on the spot always get ripped off. Odds are good you won't even be informed of anything until well after the funeral is over unless there is someone in the family you can trust to get you the information. Either that or you need to find out who is going to be executor of the will, pray its a lawyer, and get that lawyer's information and attention before anything goes down.
But looking at your post, I am going to guess you won't have the strength of will to get the information you need.
As far as dual citizenship goes, neither country does that officially. What that means is that its technically not allowed, but since they overlook it or don't have the means to enforce it, it is possible in the real world. If they ask for your passport, you just give it to them. Go to your other embassy and get a new one. Of course this requires a lot of lying, like saying you renounce one citizenship and not mean it all, like Chris Savoie. But at this stage, I would be surprised if Japan would give you a passport, and if they won't, you aren't a citizen.