You acknowledge the likelihood that the child in question is the result of your involvement with this woman - good for you.
The highroad suggests that you do all you can to give this child the best possible education and opportunities later in life. The last thing anyone needs is another struggling Dominican feeling compelled to attach themselves to one or more foreign tourist income streams.
As for this particular situation. You have probably managed to progress through this relationship without getting documented as the birth father. Assuming that you are the birth father, apart from your moral obligations it may be that currently you have zero assignable legal liability for this child until paternity can be proven - at which point the game changes.
From what you have said, this to me sounds like an attempt to come up with a compelling reason to convince you to get documented. The women may not have accurate information that she can get all ID documents herself and has been told that she needs you for some or all of this process. It may be that whatever EU country she may wish to relocate to is requiring the details for the father of the child. It may be none of these reasons and something else entirely. I suspect her new European wallet is not willing to take ongoing legal responsibility for the child so mom is attempting to correct the omissions of the past.
Right now, you may be in the most advantageous position that you could be in. Legally, this isn't your child and you do not have to pay mom & child support for the next 18 years. Morally this is not the case but we'll keep this focuses on legality for now. Anything you do from this point forward could make your situation less desirable.
It's always nice to know if a child is really yours. In this country, little can be taken for granted or at face value. However, the ambiguity of not knowing for sure, can work in your favor. If paternity is established, then you cannot question/challenge it later on. Unless you absolutely have to know, I wouldn't get a paternity test until forced to or the circumstances change and there is a clear advantage to you and the child to do so.
I would not return to the DR for any reason ever. You may not know the extent of the allegations this woman can make against you and it won't be until you set off alarms at the airport here that you find out.
You have a choice:
1) Status Quo: Support the mother & child from afar with as much fairness as possible until she shacks up with another foreigner at which time you may focus your support on the child - not that anything you give will be put aside for the child.
2) All In: Get documented, assume your parental rights/duties get this child out of the DR and into a country where she can thrive and mature into someone other than yet another potential honey pot. This will probably involve a lot of shared time with Mom too, but in for a penny, in for a pound.
3) Walk away: You could suggest that Mom get her current European Beau to assume legal responsibility for the child (adopt) and then let him deal with all the paperwork of getting the child into Europe. I would also not facilitate US citizenship for this child until she is 18 and considered an adult and then only if the daughter specifically asks. If daughter then wishes to help mom get into the US to stalk you for past affronts, that would be the daughter's decision. Mom being the guardian of record for a minor child with US citizenship and an absent US Dad, could open you up to a whole new world of enforceable legal liability.
This is something you have to figure out for yourself based on your own circumstances and wishes. The preponderance of the suggestions here seem to suggest that returning to the DR is not a good idea. Getting involved in the documentation of this child seems to suggest that it is you who is being documented and since you have not been a part of the lives of these two (there is in fact someone onsite now) you may wish to leave it that way.
Good luck. In the end, remember on some level, this child's future depends on you. Start a college fund in a US bank, name the child in your will, and prepare for the day when you can make a major difference in this child's life for the better regardless of how you feel about Mon now or in the future. Don't forget to take whatever steps are needed to facilitate that child getting US citizenship on her own, should she choose to do so, after she turns 18 and you are not available because you have been run over by a bus years earlier.