A similar situation happened to me last year when i was traveling with my daughter to Colombia. What happened to this woman was a carbon copy. Her European passport. Ah overstay fee. No she was born in DR. Shown DR passport. Ah permission letter... * But I knew my rights and the law and insisted that he was wrong. He tried to call supervisor but at 6 am on Sunday morning... (you can imagine). Finally he let us go. Upon return I reconfirmed with supervisor that "my side" was right and that officer was wrong. That was last year June. When I traveled with her in December to Europe I called SD office to reconfirm nothing had changed, and also went to the airport to speak to the supervisor (no need for permission letter if traveling on dual nationality foreign passport, and no overstay fee if dual nationality and DR nationality can be proven). Even got one officer's whatsapp number (quite cute) to contact her if there is a problem and the officer on duty that day would give me a hassle.
The immigration officer was WRONG and she should have known better. The law is on her side, and every time I travel I make sure that I check with immigration in SD the law (or policy) had not changed.
Since this year they again give the blue slips when you enter with the child on foreign passport, so that is again the "security insurance" for cases like this. Blue slip is a get-out-of-jail free card (well, get out of the immigration counter to board the flight) as that one by default removes the need for the Permission letter.
*
http://dr1.com/forums/showthread.php/151404-New-exit-procedure-leaving-with-minors
What you guys need to do in situation like this, is call SD office, reconfirm no need for permission letter, get the name of the person you speak with, write it down along with the extension number, and if faced at the airport with the issue, use this information, insist on speaking with supervisor if necessary, and I am sure the supervisor would not want to be dragged into the office of his boss for not complying with the official rule/procedure. Just be respectful, but insistent on your rights.