I'm no expert on this, but I fish with a guy who is. It is entirely possible to fly fish for billfish from the same boats used for sport fishing tackle. They put the same downriggers out, etc. and use a teaser line on conventional tackle to bring the fish within casting range of the stern. It's timing, but the guy on the teaser pulls the teaser at the same time the fly angler casts the (huge) fly right in front of the fish. Apparently, it works really well.

They also chum up a lot of deeper bluewater fish over reefs so they can be caught on flies. And Dorado are routinely caught on flies. You may want to do some asking around with regard to fly fishing friendly charter captains down there. Just be wary and don't jump on the first one who says they can do it. Get referrals. Some great sources of info for bluewater fly fishing tips and guide recommendations are:

The guys at TFO HQ. Call them, they won't mind one bit. Tell them Ken Morrow told you to call.
Dan Blanton out in California is HUGE into that stuff. He has a whole website devoted to it. Google it up. He'll talk to you, too. He's on Facebook.
Black Fly Outfitters in Jacksonville, Florida. Give them a call.
Captain Al White of Boca on the Fly is my friend I mentioned. He wouldn't mind talking to you, but I don't know if he's familiar with the Dominican R. He could give you some pointers about finding a good guide though.

That should give you enough leads to get educated enough to have a great trip. Best of luck to you!