Why doesn't Core release an update that includes Segwit AND a block size increase? It seems that this is the only thing that both sides would eventually cave to and agree on, that way companies needing Segwit will be able to do business and we can get a larger block size. With such a division being created, it's going to be hard to get an agreement at all unless there's something met in the middle.

Also, if Segwit works, which they think it does, then increasing the block size won't matter because not all the block size will be used. So what's the prob?

One of the arguments is that miners will get a higher fee with smaller blocks. If the miners are wanting larger blocks with BU, obviously they don't care about getting higher fees.