Aside from losing the ability to remove some/most of the break material and extraneous solids that settle in the chilled wort (kettle), I cannot foresee a negative impact from leaving your false bottom in the fermentation vessel (it's SS after all). Some folks toss in everything from the kettle anyway, in which case your scenario would be very similar. Of course, it would seem very easy to remove the false bottom at some point during the boil just so it doesn't have to sit in there for the duration of the fermentation (it's 6s though).