There should be some training about administering the vaccine appropriate (mRNA sequence) for the different races of this is true. Otherwise I'd expect the nurse administering to shrug and give her the vaccine.

Also do you think they would have pharmacist assistance doling out these jabs if they had to rely on them to pick the correct lot number for the correct race?

If they have made this vax sophisticated enough to require certain formulations be given to certain races, then I'm really surprised nobody said anything already. Perhaps the money for the people administering the jabs is that damn good that they happily look the other way and kill off their fellow man within the community.