Basically the widest I can get
I don't want to be overly critical of you choice, but are you looking for offroad performance? the thing with wide tires is that they usually end up weighing more. More weight and offroad = no good on a patriot.
To be perfectly honest other aspects of the MK are going to limit you before say the difference in width of a 235 tire vs 255 (just as an example)
AIC stuffed 245/65/17 in there with hardly any room to spare. I"ve wheeled with both 235/65/17, 225/75/16 and by accident (got into a dumb situation) with 225/60/18 Tread pattern and weight is far more important than the width. My current 235/65/17 tires weigh 34lbs, stock SRA's weigh 28lbs, so I have an increase in weight. When I had my 225/75/16 tires that weighed 40lbs+weight of wheels, it was noticeable and really hurt me offroad.
Now if you are going for looks as well, nothing wrong with that but be aware that you will have heavy tires, your acceleration is going to suffer as well as performance all around.
To get the sticking out past the fender look, wheels with a zero offset are going to do that or, factory steelies with wheel spacers. Also take a look at the 2013+ Jeep Compass 16" wheels, they are 16"(aluminum) and are similar to what are on the Liberty KK Jeeps.
I would look at 235/70/16, or 245/70/16 as a few members in the past have ran that size.