My back door is a double door, except that the one half is for decoration and doesn't open. My threshold is broken on the side that opens. I was wondering if I could simply cut my current threshold, which is about 6 feet long and replace the broken part, by hacksawing the length I needed. Do you think this will work? I just don't want to have to take both doors out, which would likely mean I would have to replace the beam splitting the two doors.

I don't see why your idea would not work. There is no sense in replacing the whole thing when the other door is stationary.