I would try to break out the Nitrox portion. It is a one night program where the student shows up with the Knowledge Reviews complete. Figure a long weekend to do the work. Three hours for the classroom, test and gas analysis portion.
You might be able to find an instructor that will teach the course to a single student. Ask the LDS. Sure, it's going to cost a little more.
PADI might be able to do the card that fast. First, your brother can mail the PIC card using any method he wishes, there is no requirement that the instructor mail the card. Overnight is good. PADI is pretty quick with processing (so I believe having worked with PADI on another matter) and will probably ship the card in 3 or 4 days. Then it's snail mail back to your brother.
Second, your brother will have the temporary card that is given at the end of the class. You can check with the resort but there should be no problem with them honoring the temporary card. Well, I guess there could be but there isn't around here.
Call PADI and see what they say. They have expedited handling on replacement cards, there may be a plan for new cards as well. Yes, it is at additional cost.
Which shop are you using?
Richard