First, PLEASE ask your employer for a schooling allowance so that the private schools, and even international schools, are not too much of a financial strain.
I can also recommend Waldorf schools. Speaking as the parent who oversees homework and school projects, it helps a lot that Waldorf school curriculum is standard world over. What that means is in third grade, they are reading the same stories in the U.S. Waldorf schools as they are in German Waldorf Schools.
My German is okay, but when I explain concepts to my kid, I do so in English. It is fairly easy to obtain Waldorf curriculum information online in the States, as homeschooling is quite common back home (I am also from the States).
My advice would be to come to Germany now, before schooling starts for your kids. I think it would be way easier to transition back to education system in the States later on in their lives than to begin in the middle of German grade school, if you are inclined to go for the public system here (I don't recommend it - No offense, as you say you are native German)
Believe it or not, when it comes to the schooling system, your native status might make the teachers and school officials more inflexible. I am an American married to a German. When our daughter began grade school, it appeared we were dealt with more harshly than the few non-native expats that bothered with the public schools. I suspect, but cannot prove, that underlying the laxity stems from this assumption of ---"Well, they will be leaving in a few years to go back to Australia, U.S. India, etc., so we don't need to make an issue of every little thing with the kid." Your being German might prompt them to scrutinize your children from the get-go whether they believe "little Matt" is hauptschule, realschule, or gymnasium material.
There is a reason expats tend to avoid the public schools here.
What is most important is that whatever school path you choose, your child does not wind up with FEWER options in life than he/she otherwise would have had just because you moved to Germany. Good luck!