Copa's flight prices out of SDQ are very competitive indeed compared with Punta Cana so one hopes that they don't charge such high prices for a PoP flight.
You have to realize PUJ is a tourist destination, with heavy demand. Demand is much higher to PUJ than to SDQ and STI combined. Latin American tour operators book the cheapest fare brackets months' in advance. As time progresses, only more expensive fare brackets are left available for booking. Having said that, you can usually get competitively priced flights out of PUJ (almost same price as SDQ) if you book 6-8 weeks in advance. But for last minute travel with Copa, SDQ is cheaper, simply because the cheapest fare brackets get booked to/from SDQ at later stages than they are getting booked to/from PUJ.
P.S. DR has a very liberal aviation policy, if Copa is interested, DR will grant them rights. It all depends on route planning at COPA. And when they start, they will have most likely prices like STI, which is a bit more pricy than SDQ (they compete with Avianca on both SDQ and PUj routes).
In the news from yesterday, LATAM will open up base in Bogota for flights between Caribbean and South America to directly compete with COPA. Let's hope DR will be some of their next destinations (they already fly from Bogota to Cancun and Aruba), hopefully they add up flights from BOG soon to DR (SDQ/PUJ), that will put more competitive pressure on Copa's fares (Avianca's routes to DR started exerting downward pressure on COPA's fares about a year or so ago). LATAM already flies to PUJ from LIM, EZE and SCL, but BOG would actually be a similar proposition as PTY - hub to connect Central, South America and Caribbean.
http://www.reportur.com/colombia/2015/07/08/latam-elige-bogota-para-abrir-su-hub-al-caribe/