This really sounds like a wiring issue. Trailer wiring routinely gets damaged, bad grounds, etc.
I would try hooking up to another trailer OR try your trailer on another car to see if the problem persists.
Are you are using a 'powered' trailer light converter? It sound to me like you aren't. Powered means it has a separate 12V power connection from the battery:
Image courtesy Professional Wiring Products
Read more in these threads:
zer...51101http://forums.nicoclub.com/zerothread/333642
If you are using an un-powered converter (no external power connection) or just a simple harness, you are asking for trouble. The Rogue's lighting system was never designed to handle the extra power needed to run trailer lights. If you use a proper powered lighting converter, you should never have problems with the turn signals in your Rogue... any problems would only be with the trailer.
As for upgrading the flasher relay. This is not possible. The turn signals are integrated into the Body Control Module... basically a computer that runs the basic controls. ie turn signals, headlights, intelligent key, alarm system, etc. It's a sealed unit, there is no relay to replace/upgrade.
Go any buy a proper trailer light converter if you do not already have one.