Well we are talking about a period of two-hundred years before they actually merge, so I think that's enough time to become close enough.
I realize it's difficult to do, but it's not impossible.
1600s-1800s? Still not really going to happen. With the tech and population levels early on, you're not really going to see too much cultural diffusion. You're talking 25-30 million Japanese, versus the couple hundred Dutch that are able to make it over. You're going to need to ship a hell of a lot more Dutch over for Japan to start
really feeling cultural influences, and that's just not going to happen.
1) Not enough Dutch. The population of the Netherlands was something like 1.5 million at the time. Of those, how many are going to be willing to make the arduous journey to Japan?
2) The tech was shitty. Of the people who are willing to make the journey, how many are going to survive it? When the first Dutchmen arrived, there were twenty survivors out of 500. That's 4%. So even if the entire population of the Netherlands decided at once to move to Japan, we'd still see only 60,000 Dutch in the country. That should be enough to make a cultural impact, but I don't think enough to completely convert the country into wanting a union with the Dutch. But that's moot, anyway, since it's not like the entire population of the Netherlands is going to want to head over there. You're not going to have 60,000 Dutchmen. You're going to have a few thousand at most, many of whom are going to be traders who don't spend long enough in the country to really exchange cultures.
3) So, say you fix the first two problems. Tweak some numbers here, have some ASBs there, and you have a few million Dutchmen in Japan. But one more problem. Japan at the time was not known for loving Christians. There was constant, brutal repression of the religion. If it starts getting too powerful, a Shinto or Buddhist shogun will put it down. The Japanese were very effective at keeping Christianity down historically. It still existed, but was practiced in secret. It certainly wasn't part of mainstream culture. To fix this, you'd really need to have a Christian ruler, or at least a ruler with a lot of sympathies towards Christianity. This becomes difficult, because a Christian shogun would be very unpopular with daimyos.
4) Logistics. Even if Japan decided to have an Imperial Union, how is that going to work out well? It'll take months to transfer information from one to the other. A colony system works, where you have a local governor who is relatively empowered and independent. Having Japan be a province of the Netherlands just like any other would not. You'd need to make Japan a colony, or at least a colony in all but name. Since there's not enough Dutch to dominate Japan like that, you'd have a province that basically rules itself, and Japan would end up almost completely detached from the Dutch crown. Since Japan rules itself, what's the point of the union?