Although they share the same corporate owners, BP in the US is really not as closely associated to BP in the DR as you might think. Different rules and regulations in the US and the DR. The same situation is true with Citibank in the US and DR, and Scotia Bank in the DR and Canada.
For example, to get money from BP in the US to BP in the DR, you would have to do a wire transfer or deposit a check from BP in the US to a US dollar account in the DR. No different than using any other bank in the US. Because there is no easy way to connect the money between the US and the DR, it is probably better to use Washington Mutual or BofI (Bank of Internet) in the US for their lower fees than most other banks in the US.
Additional info and a request
I have found using my US bankcards to withdraw pesos in the DR does not provide a great exchange rate. After calling the banks, MasterCard and Visa, none of them provide a good explanation as to why the rate is not better. One card normally provides a better exchange rate than another, and neither company has been able to explain the difference.
I have yet to find someone in the DR north coast that will give me a rate on a personal US check that is as good as the rate for US cash.
If someone has a better solution, please post it.