No.. any cheap RD will function .. 6 speeds refers to the gears not the RD, it just shoves the chain sideways ..
the big cog has to be smaller to let you use a shorter cage RD.. lose the whole wheel get a 3 speed IGH and then all you cope with is a chain tensioner ..
but that wont be cheap .
have you tried to find the minimum chain length that works? then the bottom pulley wont stick down so much.