I'm considering new wheels on my 2010 Fuzion TE II, factory wheels are 7" wide, new wheels are 6". Would there be any issues I could expect running a 1" narrower wheel?

That would mean using a narrower tire that more than likely has a lower weight rating.

Why new wheels? What tire came on the trailer from the factory, and why do you feel you need to change sizes?

I would not go to a narrower wheel, same size tire will not wear correctly and smaller tire may not handle the weight.You need to consult a "real" tire/wheel expert.

Check the specs on the tires you have now...that will tell you what the rim width range is. For ex. a 235/80-16 Goodyear Marathon has approved rim widths from 6.0-7.5"