Is a MOSFET's body diode different than an ordinary silicon diode at audio frequencies?
I believe you are not a sound subjectivist - a person who insists he can hear essential, but unmeasurable sound quality quantities, for example "warmth" or "depth" as soon as he sees a vacuum tube which actually has only the filament connected, a couple of extra zeros in the price label, etc...

Of course the distortion characteristics of that clipping circuit depends on the diodes. The static If/Vf  curve differences are not the whole truth. I believe (haven't measured) that longer storage time (see NOTE 1) causes treble loss when compared to a fast diode. It can be good  in some cases and something untolerable in other cases.

Very radical clipping characteristics change is possible by adding one resistor in series with parallel diodes. That can be even worth a new potentiometer. Adding a resistor in series with only one diode makes the clipping non-symmetric and turns the device to a fuzz effect.

I do not know the storage times of the suggested diodes nor mosfet diodes and definitely cannot tell the differences as sound description words. You must build a test circuit. A signal generator and an oscilloscope will help much in analyzing, but a proper musician is needed to judge the value for music. Get the musician after you have found clearly measurable differences.

NOTE1: Minority carriers around a P-N junction can live in semiconductor materials a long time after the forward voltage has been reversed. This is the essence of the slowness of a diode. Some diode datasheets call it "storage time" There is also "reverse recovery time" which is the time needed to build the depletion region, but that is generally shorter in PN diodes. 

Diodes which are intended to be used as 50 or 60 Hz high current AC rectifiers in power supplies are often useless in


high frequency circuits because they seem to conduct as well to both directions
pulse circuits because every time the voltage is turned to reversed, a hefty current pulse is conducted to the unwanted direction


BTW there seems to be no easy to find datasheet available for diode MA 150.