Can you determine whether the bruises were due to the "lousy poking" of needles when he tried to take the blood samples ... or were due to the "squeezing" with his hands while he was trying to hold your baby down?
If you're not sure, then take a look at the bruises ... if it's caused by the needles traumatising the blood vessels, the bruises should be circular in nature or even like a cloud/flower & the colour should be fading outwards ... if it's caused by his hands exerting force, then you might see a shape of his hands (or just a part of his hands) & the colour has no consistency.
Should you be able to determine that it was caused by the squeezing, you can take photos & proceed to your neighbourhood police post to make a report ... before filing a complaint to MOE with all the details attached. You probably will not get any monetary compensation (unless you engage a lawyer to pursue the matter in this direction) ... but you'd definitely get a proper explanation from the authorities, including an apology if it is deemed obligatory.
However, if it's due to the incompetent needle poking, then there isn't much to pursue ... I mean, if hospital staff (including the less experienced nurses & trainee doctors) also can get it wrong on the adults, what's stopping them from saying your baby's blood vessels are even much smaller & thus, even harder to work with.