FIR filters still have a small transition band in order to have an inaudible passband. So you never will get a completely instant cutoff - even with a FIR filter. How steep they are depends on the number of taps they are using and other things. The more taps, the slower the thing will be. The steep FIR in Renoise’s multiband uses 65 taps with uses a kaiser windowed sinc window, which already is a good compromise between speed, loudness of the stop band and steepness.
In your example you can simply set the lower frequency a little bit higher to compensate for this.
Note that when checking the “steepness” in the spectrum analyzer, set frequency to linear. Else it will look like it has a different steep at the left side of the passband.