False bottom is only useful if you are using leaf hops. Create a good whirlpool, allow to settle, open the valve just enough for flow, and I'm sure the diverter will do well.
I like to separate out the trub for two reasons. 1 - I don't need to waste space in the fermeneter. 2 - way easier to save yeast cakes when there isn't twice as much mucky garbage than yeast.