A.C.
I'm a phlebotomist and yes a lab is going to be a little more prepared for dealing with a little one. One of THE most important things for you to do as a mom is STAY CALM. Yes it's going to be uncomfortable for little man. BUT if you stay calm, try to distract him (with silly songs or poems or something like that) and HOLD HIS ARM STILL it'll go much easier.
The other option is to go straight for the papoose board. And it's designed like it sounds. It's confines and restrains a small child so that they can't grab at the needle and keeps their arm still. I personally prefer a parent to hold them as often as possible. I think it's more comforting for the child. But I'm also not averse to using the board if mom can't or won't hold kiddo still.
I'm also a mother of 3 who have all had to be held down for one reason or another. So I understand how mom feels too :)