I have heard of people just using regular cheese from the store. Obviously it just gets more runny. I would think it would be fine for brats where you don’t remove the casing but may make a mess inside the casing of summer sausage.
If I am making something like summer sausage or snack stix I use high melt cheese because it holds its shape and looks visually appealing when it is sliced. It also has a nice texture.
If I am making fresh cheese brats I tend to use a good sharp cheddar cheese for the flavor. Shredded cheddar will add flavor but you wont have the texture like a diced high temp cheese. I use sharp cheddar because it seems to be less greasy when the brats are cooked.
I have also used swiss cheese but the the rest of the family doesn't care for swiss all that much.
I've always purchased the high temp cheese when making Venison Summer Sausage. This year I made a batch with regular cheddar from the grocery. Worst part was chopping up the 3 pounds into 1/4" cubes. It worked well in the sausage but it is kind of runny when cutting up the sausage as others have mentioned. Taste wise it is great. I did not allow my smoker to go above 180F.
You are going to need high temp cheese.
I tried making summer sausage with some expensive aged cheese from a cheese expert in Wisconsin that was thought to hold up. It didn't, the result was hollow spots in the sausage, where the cheese was. Wasted money, bad batch of sausage.