This one can be tedious and a bit tricky, as the title says use each weapon type to kill a demon in battlemode. For the BFG the rounds will need to go to i think 2-0 or 1-1 to allow you to gain access to the BFG at the interval where you select upgrade, there's no other way to get the BFG.
Also note that when using the BFG make sure the demon is at its lowest health possible to be sure of the kill and be aware that a direct hit with the BFG will not count for this, you need to kill them from the beam itself.
If you find this one difficult you can do this in a private match too ideally with 3 people total.