This is what we mean when we say something falls between the cracks. Sorry for that.
First off, in the United States, family law is not federal, it is individual to each state. This means there is no federal agency or official charged with enforcing family law. When it comes to marriage and divorce, the federal government and their border agents are after those seeking to gain benefit through fraud (i.e., attempting to attain status through fraudulent marriage). Likewise, they don't have formal cross-jurisdictional protocols controlling the right of the US government to demand personal, private family law records from other governments. Nor do they have the means to pursue that. What this means, in practical terms, is that no family law judge sitting in the US has authority, interest or means in international marriage and divorce beyond those cases initiated in their court. I hope you read that slowly and carefully. Ask me if you need clarification.
I suspect that you have a very specific cause in mind that necessitates you being in possession of a divorce decree. And I get the impression that you and your ex-spouse are in agreement and working together to solve this. You might find it valuable, at this point, to take a step back a little further into history.
Let's consider your marriage. You two know you got married. I suspect that your friends, loved ones, and hopefully family know you got married. Then there is that clerk somewhere in Denmark (who records more than 100 marriages every week) who knows you got married. Theoretically. Possibly. Maybe. You've been residing in Russia for some years now, and the Russian government doesn't know you are married. Marriage is illegal there. So, in actuality the Russian government, if asked by some official of some other government, can only state with truth and authority that it is certain that you are NOT married.
At the same time, some US federal official decides to investigate the most highly unlikely case ever. Someone is trying to gain entry, not by claiming marriage but by hiding a marriage. (ridiculous!) So, with a budget of zero, and the authority to match, goes from country to country demanding that they open up the (extremely) private records of family courts in search of the evidence he needs. When they ask him to demonstrate cause, he boldly tells them that his sharp mind is cause enough. They agree and give him cups of tea as he searches through the private affairs of their residents. It is never-ending, but he is proud to be working to stop the flood of unwanted divorced persons trying to gain entry to the USA. - - - - You get the picture.
Also, consider the option of an international divorce. I would provide a reference here, but a simple Google search will yield many providers. It is expensive and time-consuming, but an available option. I'd rank it last. There are also varied laws by jurisdiction - internationally. I'm not expert enough to know of any jurisdictions that might not require extended residency. I suspect some won't. I am confident, on the other hand, that there are several that you might access with relative short windows of required residency. This Wikipedia article should give you a good start.
You are also maybe a little fast in dismissing Denmark as a possible venue. European Union states, at one time, experienced a tangled mess regarding family law across jurisdictions and specifically divorce. In 2003, however, the EU implemented Regulation 2201/2003 providing for conferred jurisdictional competence by way of exception in cases involving applicants for divorce similarly situated to you. And now jurisprudence, in a wonderful show, is beginning to catch up with regulation and public sentiment on the matter. The provisions seem complicated, but they are definitely outside the abilities of a couple of Russians who have managed to get this far. Here's a starting point to get the ball rolling.
Aside from those three solutions, you might be stuck. You would be ill-advised to seek some creative solution outside of the law. And I struggle to think of anything that might work. Unfortunately, justice and law are often connected by a mere thread, and sometimes not at all. You are left to forge ahead on behalf of others, who enjoy the fruits of your labour. For that, you have our gratitude. And while I am no expert on the fight for justice, There is this man - an American, who is. He wasn't gay, so much as black. Be he figured out a lot regarding justice.
There are two types of laws: just and unjust. I would be the first to advocate obeying just laws. One has not only a legal but a moral responsibility to obey just laws. Conversely, one has a moral responsibility to disobey unjust laws. I would agree with St. Augustine that "an unjust law is no law at all."
-Martin Luther King Jr.
I salute you.