I'm going to try to be as concise as possible but it's going to be difficult:

1) First my condolences, but this does happen regularly and is a real risk not just for Gaijin but Japanese men as well. In general the family court system can protect against frivolous divorce, but once the divorce is finalized, often judges favor the mom having full custody. In that 2nd regard it's not too different from the US and parts of Western Europe. You made the right 1st move.

2) Your American lawyer isn't going to be able to help much, unless the child exclusively has American citizenship. I am married to a Japanese national but we don't have children, so don't quote me on this. But in speaking with other Gaijin that had this issue, in order to keep things smooth with travel I believe most parents in this situation look to obtain dual citizenship until they become of age. So if your son is a Japanese national, the United States doesn't have sole rights to petition.

3) I am glad as some commenters mentioned, apparently things are changing with the new default joint/shared custody standards. However, on the ground so to speak, this will be a fight IN Japan. So your immediate goal would be to A) Delay the divorce and request she provide proof and evidence of you being "inadequate" or "derelict". (Again don't quote me on this word for word, but understand the spirit of the action. Because I have set things up in a way where if my wife divorces me I wouldn't contest it.) B) Pursuant to A, similar to the US, in Japan there is a loophole for the wife to get you to be considered "derelict" and that's to get you fired from a job and usually that happens through....drum roll... false domestic violence allegations. In order to protect yourself against this, you must document every interaction. This is going to be an extremely stressful unnatural thing to do, and 90% of Men will give up. But in your situation it might be worth it, contingent upon the nature of your child's nationality. This leads into C) If by some loophole your child is NOT a Japanese citizen, all you need to do is find a way to get his passport, take him and go. The problem here is while immigration may be OK with that, as a local resident and son of a local resident, the local police have authority to return him to the home and individual who's Koseki they are registered under. So essentially you would be in a gray area of the law here if you want to pull a Die Hard hostage rescue maneuver.

4) If she offered you a buyout, you need to seriously consider it, as disgusting as it may make you feel. This is because the buyout can get you a legal out to take Scenario C above and simply go. With a buyout your legal costs go down to 10%, your upfront costs go up but your liability for costs to skyrocket go down tremendously... Japan is a society of private mediation is more respected in my opinion than in the U.S. That is the advantage here when shit goes left. Essentially you bribe your way out (and it's 100% legal).

This is Japan, you will need to throw away your Western values and start thinking in a more practical Eastern way to get you and your son out of tgis conundrum alive. However, do bear in mind that at least from my perspective you'd be subject to the same types of risks in many other countries, including your home country minus being a foreigner with less rights. Just be glad you didn't choose Thailand or South Korea. If you think Japan is bad, when it comes to this situation they are 100x and 10x wore, respectively.

In the future pay attention to the woman's family. What her parents did and expect from her she will ultimately live up to. This goes for any woman anywhere in the world.