I was curious where you found information about Mateus being a low population server because I hadn't seen anything about server populations since August of 2013. Then I found this site - http://xivsoul.com/ which counts server population based on how many people are in a free company on the server. Not a perfect count but the percentages between servers are probably good.
From that it looks like Mateus has the second to lowest population among the NA servers with about 75% of the population of most of the other servers and 55% of the most populated.

That being said, on Malboro the time of day has a tremendous effect on the number of people logged in. I assume it is the same for other servers. Malboro has a lot of east coast US players. After midnight Pacific time, most of the population has dropped off. I don't know what times you are playing but that may also be an issue. Some people who work odd hours in the US have had better luck on the EU servers because of the time difference.
It may also be that there aren't a lot of new players on Mateus and everyone is clustered in the higher level areas.

The duty finder is across all worlds in your data center, by the way. So the world itself having a lower population shouldn't affect your use of the duty finder. I realize it does affect your experience with the rest of the game however, including free companies, FATEs, and forming pre-made parties.

Also, almost all MMOs require you to pick a server for your character as far as I know. So it's not that odd. World Of Warcraft charges $25 USD to change servers.

Regarding leveling, everyone plays at their own pace. We have several players who all started with the PS4 release. Their levels all range from 18-44. I think one of them is actually 50. So while re-leveling to 27 may be nothing for some, it is a significant investment for others. It depends on how you play.

So, for you the question is which is going to cause you the least frustration: paying the $18 for the server transfer, continuing to play in a world you feel is empty, or re-leveling?
Personally I would create a new character and re-level them. But I have the subscription that allows multiple characters. If that is what you have, I'd create new characters in several worlds just to see what the online population looks like during your play times.