The very simple answer is that in English, as you will have guessed, it sounds a little strange to say "native Japanese".
It's that simple: find another form.
In, say, Canada, the USA or Australia, you have the "native" people (before the Europeans arrived and killed most of them), and you also have the currently dominant Western-society populations.
Generally (but not always) if you say "Native [Australian, etc.]", that generally means an indigenous person, rather than the genocidal arrivals of the colonial era.
On the other hand, you might sometimes say "I'm native to New York", meaning "I was born there" as opposed to being one of the very many people who move to NY to live there.
This particularly applies in very unusual cases like, say, Monaco, where there are only a handful of true native Monégasques and the rest are visa-holding residents.
Note that in both these cases it does not really apply to you.
So your question .. If this expression can mean both mare Japanese people and Japanese indigenous [Ainu etc.?] people, how can one distinguish between those two? .. indeed, the answer is -- you can't. It tends to mean the latter more than the former.
So, generally in English, saying "native .." and then a country such as: Japan, France, etc. makes less sense. It's that simple.
"Native country" tends to apply more only to the New World countries where the Europeans exterminated -- or nearly exterminated -- the native people.
If you do say "native French person", that can sound almost a touch political: at the moment in European countries, there is a tremendous influx of new residents (from Muslim countries, Africa, etc.).
Even within Europe, in say Britain there is a big influx of Polish people. (Thank God, so we get some good food and table manners there! :) ) So if you say "native British person" it CAN TEND TO MEAN a "real" British person, not one of these "newly arrived economic immigrants".
So that's a little fraught with danger.
There is SOME CHANCE that if you say "native Japanese" it sounds like you're saying: "I'm not one of these damned Koreans or Brazilians who move to Tokyo to run a restaurant!"
Now to further add to the confusion, most people would be aware that in Japan you do NOT, generally, have a lot of economic immigrants (only from Brazil, right!), so this would further confuse people.
1) Native __ of New World country __ tends to mean one of the remaining indigenous people not wiped out when the Europeans arrived.
2) Native __ of some country like France, etc. __ -- at the moment, in current affairs -- tends to mean "not one of the many new economic immigrant arrivals". This can sound a little bit racist or political, so take care.
3) "Native of __ city like New York, Monaco __ in very vibrant, mobile cities is like (2), but has no racist or political overtones
4) In your case, listeners will realise that Japan is in category (2) (like France, not like the USA), BUT Japan is known for having unusually few economic immigrants (unlike, say, Holland or Britain, where it's a hot current affairs topic), so it would be additionally confusing to say that.
Note that as you point out, separately, in the phrase "native speaker...", that simply refers to your mother tongue, and has nothing to do with your race, birthplace etc., or the other issues raised here.
My very short answer: it's completely clear if you say "I'm a native Japanese speaker." However in short I just would not say "I'm a native Japanese", as it's plain confusing!
Say you also speak great English, and on the phone (say) someone was confused about where you are from; if I was you I'd simply say "Yeah, actually I am Japanese" or "Actually, I am from Japan", or even expand: "Sure, I'm Japanese, I was born and raised in Japan, I just speak English also."
There's a "short" answer for you! :) Cheers...