How can I obtain a three-month visa for Thailand upon arrival with a one-month visa?

yeah more strange is that they have no consulate nor ambassy there, it seems they are in bkk !

here :

pretty strange how the list the thai consulate in Budapest Hungary as the place people from the DR go to get a thai tourist visa :O

It doesn't seem Dominican Republic is on the list of countries that are either Visa Exempt or Visa on Arrival which means you cannot enter Thailand unless you get a Visa in advance from a Thai consulate or embassy, according to the Wikipedia list. Once you get either a single entry or multi-entry tourist visa at a consulate or embassy, you can stay 60 days, but can extend it at the immigration office for either 7 or 30 more days for 1900 baht (depending on what country you are from). I can't find whether Dominican Republic is allowed a 7 day or 30 day extension.