So, if anyone was curious. No, they did not fit.
The mounting studs are too far apart to fit into the holes on the car and the cutout where the mirror inserts is not the same. I've also found that I bought the wrong mirror as the ones that I took off of my car are heated and the compass ones that I bought were not. My mistake. The mirrors would not turn when plugged into the car and I assume that's because of the power difference between the heated and non-heated mirrors. If you were to buy heated mirrors to replace the stock ones on the patriot then I assume they would rotate and could be adjusted.
I noticed that the mounting plate that holds the studs on the compass mirror is able to be removed while the one on the patriot mirror is not. My thought would be to take off the mounting plate from the patriot mirror and find a way to transfer it onto the compass one so everything would line up correctly. I'm not 100% sure about this but I believe that the patriot mirror is entirely one piece and the mounting studs can not be moved to the other mirror, but please correct me if I'm wrong. It isn't going to happen today since I bought the wrong mirror anyway. If anyone has any ideas on how I could make this fit then that would be greatly appreciated.
Here is a picture of the mirrors side by side, note the distance between the studs.