The first thing to note is that fashions changed rapidly in ancient times, just like they do today and one "Phoenician" might be wearing something completely different than another one. Also, a foreigner who was doing business in Rome normally would dress just like the Romans. Wearing foreign garb in Augustan Rome would not be a recipe for success. Also, remember that Egypt was already a satellite of Rome at that time, and the leading citizens in places like Alexandria were imitating the fashions of Rome.
If you really want to show a different, regional style of dress for Egypt and Tyre, your best bet is to go with Greek styles. Both Egypt and Tyre were run by the Greeks and the traditional outfits were Greek. There is a very famous mosaic called the "Nile Mosaic of Palaestrina" which depicts a range of views of Alexandrian life (as imagined by Romans). This mosaic should give you a good idea of the types of clothing that was worn by the upper classes in both Alexandria and Tyre at that time. Here is a detail from that mosaic:
There are other Ptolemaic mosaics which you can find in archaeological books with similar content. In the mosaic above you can see men wearing both chitons and robes. Romans never wore robes, but would wear chitons casually. For your purposes, a robe like the long-sleeve ones on the right in the mosiac with the colored borders would probably be most appropriate, as that would seem like an "oriental" outfit to a Roman.