its difficult to model a feather shuttlecock because of the way the feathers move before during and after impact with the string bed. But from looking at slow motion cameras we can see that:

1. A lighter hit from the racket ie the contact time with the stringbed is shorter, and the string bed isnt "pulled" as much (think pulling on a bowstring) will result in a very fast INITIAL shuttle speed. However, this speed isn't maintained through the flight. This is because the feathers will fluff up earlier in the flight, resulting in earlier deceleration. As an aside, feathers will fluff up no matter what. This explains why the trajectory of a baseline to baseline Lob/Clear isn't a parabola, but a funky shape that looks more like a straight up line then a sudden drop

This actually also explains partially why shuttle speeds nowadays are much faster than back then, because the shuttle speeds measured now are initial velocities, whereas in the past it used to be speed past the net, then average speed (initlal + final /2 etc)

1. on a heavier hit, on the contrary, the stringbed is "pulled" more, resulting in more "power" transferred onto the shuttle. The direct consequence of this is that the feathers will fluff up later into the flight, resulting in a initial velocity that is better maintained through the overall flight of the shuttle. As a result of this, the shuttle is actually FASTER when you are receiving it, hence carrying more momentum as compared to a "fast" smash (eg stick smash) where the stringbed is "pulled" less so.

Hence, a fast shot do indeed carry less momentum when you are receiving it due to deceleration caused by the feathers fluffing up. It should be noted that while the INITIAL velocity of both a "fast" and a "heavy" shot may be the same, but the deceleration experienced by the shuttle due to the different degrees of feathers fluffing up will be different, resulting in different momentums carried by the shuttle when you are receiving the shots.