If anybody is interested, I found a work-around for making it higher resolution while searching online.

Do the texture baking in "tiles", so you end up with 4 textures that are each 1024x1024 and piece together to make one large 2048x2048. This is done by setting the UV boundaries of the baking.

Still dont' have a solution for making the baking extend beyond the UVs, though:headbang: