Well, I don't think governments of Costa Rica, Colombia and Peru would object much into the new airline servicing their markets.
We have to realize that for intra-american flights (central and south america), each country, from Nicaragua or Guatemala to Chile has a wide selection of options, TACA/LACSA, AVIANCA, COPA, and many have LAN and LAN-subsidiaries, TAM and VARIG (do they still fly?).
If Air Dominicana code-shares with TACA/LACSA and AVIANCA, that may be the way to go. Again, I don't think Peru or Colombia would object to Air Dominicana flying, the power of COPA in those countries is not that big, and if local airlines (TACA/LACSA, AVIANCA) would pursue partnership with Air Dominicana, local authorieties would be even more compliant to grant landing rights. It's also important to notice Peru and Colombia are countries that do not require Dominicans to have visa to enter (as well as Chile, Argentina and Uruguay). Now we get into the reciprocity. LAN CHILE and LAN ARGENTINA have landing rights in the DR, so DR has same reciprocity rights to land in those countries. The famous Xth freedoms. There are quite a few Dominicans on every flight to/from Argentina. DR-CAFTA may also provide for some liberation of air traffic (not sure about that) so Central America might be covered.
Mexico route might support more passangers, if the contracts are made to code-share. It depends on the elesticity of demand to pricing, would need investigation and reserach. If the demand is elastic, operating on a low-cost basis might attract traffic from Mexicans vacationing in the DR.
The next question is whether to operate full-service or low cost. Low cost is NOT that sandwich you get on board. After all, the cost of on-board refreshment is not more than 10USD per passanger on the average for a short-haul flight. It's the operations, handling, etc. that makes the airline "low cost', not the food. So Air Dominicana CAN be low cost low-fare airline, and compete with COPA.
Take this route network:
MEX, BOG, LIM, EZE, MIA, JFK, they might be able to create a "mini-hub" with the help of their code-share partners.
Also I don't subscribe to the idea "all before me failed, I am doomed to failed". In my country, in Europe, up to about 4 years ago, there were very few flighst from local airports (because other country's airports were so close by and were MAJOR airports). Several startups failed, and several got-by with problems and financial distress, then came THAT low cost airline and made a revolution. Local traffic was, obviosly, price-sensitive with high elasticity of demand, so low-cost offer attracted severalfold number of passangers than before. Now THAT low cost airline is among the ones very successful low-cost airlines. And they offer sandwich and they offer free drinks.