it really depends how much chlorine is in the water, I don't use treated water myself, but there is no smell of chlorine in our tap water
if you are concerned, warm the water up and let it stand for an hour, most of the chlorine will evaporate quite quickly (it has low evaporation temp)
that said, bathing beardies as a way to hydrate them, is generally considered a myth these days; mostly they just drink the water...but if your beardie enjoys a swim, i see no harm in it
water filers do add other chemicals to the water - we have a water softner system built into our main water supply, but the drinking water tap, is not softened as its not considered suitable for drinking otherwise (not the same thing as a brita filter tho)