Two issues here, bullet holes and blood.
Bullet holes are easy. We do not even see the full body, small weapons bullets are small, and the dark blue coat would make finding holes in it difficult, unless it was against a light colored background.
And to explain the absence of blood stains, do you not need very fine bullets. In fact, if those are less lethal, wounds by smaller bullets could even be bloodier.
Anytime realism in movies1 touches the issue, it mentions how much gorier small weapons wounds are compared with real life.
For example, is someone is shot dead, the heart stops beating and blood stops circulating. If the wound ends in the upper part of the body it won't surge; if the wound it is in the back it will slowly drip and soak the body, but the weight of the body and the clothes will prevent it from spilling around until a lot of time has passed.
In this case we have an heroic death, so we can assume the wound is on the front. If we assume instant death (a direct shot to the heart, or that when wounded the man fell and fractured his neck/head), very little blood would have been spilled. If we factor all of the clothing that the character was wearing (remember, it was winter) and that what we see is a dark blue coat (which would make blood stains harder to see than if it was just a white shirt) I do not think it would be too inaccurate.
A couple of references:
Of course, some wounds can be very bloody, specially if they hit a major vessel and the victim is not dead. And wounds by explosives/heavy weapons are a completely different thing2.
Yes, I am aware that we are talking about a Batman movie here.
2I am thinking the D-Day scene of Saving Private Ryan