The likelihood that a foreign money judgment award will be honored in a foreign jurisdiction where there was a default judgment is slim to none and the only way anyone could comment on the veracity of this statement would be to understand exactly what jurisdiction we are talking about and reviewing case law to determine if and under what circumstances such an outcome can occur.
It is also important to note, that signatories to Berne have agreed to allow access to their judicial systems in an effort to protect copyright interests from all citizens of participating countries. This, however, does not mean that they will apply US copyright law. That is, I can sue you in your own country for copyright infringement, for example, and the court may grant me access, but not under US law. They are not going to subject their own nationals to foreign statutes. Rather, they will apply the copyright law as it is in that country and in many if not most cases it will be similar, but not always.
I see no harm in filing a counter notice in general, but I am not advising you do so in your case without first consulting counsel and making sure all the details are clear. You would be smart to consult a lawyer where you are who can best advise on legal outcomes in your own jurisdiction.
DISCLAIMER: this is not intended to be specific legal advice and should not be relied upon as such. No attorney-client relationship is formed with the law firm of Natoli-Lapin, LLC on the basis of this posting.
There are too many facts about your specific situation unknown to offer good advice. Advice requires an understanding of what the allegedly copyrighted work is, who owns what rights to it, what the incentives to you and the other party are for having the material posted or removed, and details about the potential jurisdictions involved. You probably need to speak with an attorney if this is important to you.
(1) you published copyrighted material on a website,
(2) someone in the U.S. believes that material infringes its copyright and so filed a DMCA take-down notice to either the website owner or the internet service provider, and
(3) you're considering filing a DMCA counternotice to have that material re-published.
Step one is to have your own copyright attorney, in your own country, determine whether the material you've published does, in fact, infringe.
If it does, then you CANNOT file a DMCA counternotice.
If it doesn't, then step two is to determine whether the economic benefit you may gain by having the material re-published exceeds the foreseeable costs of dealing with the dispute [i.e., your time, business disruption and your attorneys' fees].
Once you work through steps one and two THEN you need to address whether it makes sends to send a counternotice. Which will require you and your attorney to weigh the pros and cons of litigating in the U.S. versus litigating in your own country. Good luck.
The above response is general information ONLY and is not legal advice, does not form an attorney-client relationship, and should NOT be relied upon to take or refrain from taking any action. I am not your attorney. You should seek the advice of competent counsel before taking any action related to your inquiry.
A U.S. court could render a judgment, but enforcing it is another thing. Sending a DMCA notice is governed by U.S. law even if the event occurs in the U.S. or under some international treaty. You could give it a shot and see what happens.