Shengen visas
Last time I traveled to the UK with mi novia via Madrid, she needed a Shengen visa even though she already had a visa for UK. She had to apply for a Shengen visa through the Spanish Embassy/consulate as UK is not a Shengen country and Shengen rules state you must have a valid visa for your country of entry into Shengen territory even if you are in transit to a non-Shengen destination.
Again this was a year or so ago and things might have changed.
The Spanish Embassy required visa applications to be filed at least two, preferably four months before the travel date. Standard procedure was filing an application, then filling out an questionnaire about the same size and weight of the SD telephone directory, attaching forms from the Applicant's bank, detailing all transactions for years past (not just a current statement), a letter from Applicant's DR employer confirming their employment, attaching copies of payslips for at least the two previous years and stating that the job/ position is held pending Applicant's return, invitation from a registered commercial institution in Shengenland, evidence of property ownership in the DR such as certified and legalised copies of property deeds and titles, mortgage contract, and the rest of a seemingly endless list. After filing the application a date will be fixed for an interview with a consular official, usually about six to eight weeks hence when the Applicant will be told if the application is accepted or rejected. If accepted and providing that Applicant passes the interview, the visa would be issued about two to four weeks later.
I managed to shorten the above procedure by persuading with difficulty, a friend who was then consular secretary at a different and incidentally non-Shengen and non-European embassy here in SD, to intervene on our behalf withy the Spanairds. My friend was reluctant at first, fearing that he would have to reciprocate, but eventually he agreed and the process was shortened to about two weeks. The Brits categorically refused to assist even though the purpose of the visit was a conference arranged by a famous British commercial institution for whose local representative mi novia worked.
Why don't you fly via the States? Silly question - there is no visa waiver for Dominicanos and getting a US visa is probably more difficult than for Shengenland.
Good Luck!!
Bryan