To make a long story short (has been discussed to death before), the only way to do this is:
Get an extension of your tourist card for 3 months, the maximum time you can drive legally in the country with a foreign license while staying legally in the country is 3 months. After 3 months you leave for 1 day, come back and do the same.
This way you are staying legally for 6 months with an interruption of one day.
Forget all the b***sh*t about international drivers licenses and getting a Dominican license in that short period of time and a foreign license being legal for more than 3 months.