Hi, man it's really difficult to find a low cost trip from PUJ to La Romana because of what you already mentioned: "hundreds, if not thousands, of people take it every day". In addition, we are in travel season.
I advise you to try with any company and pay that price before you get there without any car and you are forced to pay more. And renting a car is too expensive on that zone.
Try with private taxies, it seems like the best choice.
Good luck.