The choice of bias vs. radial tires will open a whole bag of opinions, including mine.
There is some evidence that bias ply tires, because of stiffer sidewalls, on single axle
trailers, MAY be less susceptible to sway, but there appears to be no real consensus on that.
That said, I have had both types on my several 13' FGRV's and the tire's construction didn't seem to be a factor in stability.
My Hunter has ST175/80R13 tires and tows without sway.
As far as getting larger tires, that's a\mostly a matter of wheel well clearance but remember, if you are putting your Hunter in a garage you may lose the roof clearance you need. If you want more ground clearance, swapping to a straight axle
(and adding brakes) is a better choice.