Pediatric Blood Draw in NJ?

Well, for newborns they give them a hot foot (warm pad on a foot), and then prick the heel for a few drops of blood.
An experienced nurse can do this and the baby won't even cry.
For a week or so after our son was born they tested him daily for bilirubin and as a new mom - hearing my baby cry (we had a trainee nurse for a few days) made me want to punch somebody.

I don't know how much blood they have to draw but at 15 months old - you absolutely can hold them still on your lap while they do it.
You have muscles and they don't.
Our son had pneumonia at 14 months and I had a hold where I pinned him down on the floor (towel under him) in order to get his medicine down his throat.
It wasn't easy but the doctor said that being diligent with the medicine really helped keep our son well enough so he didn't land in the hospital in an oxygen tent.
You do what you have to do keep your kid alive and " I can't " gets dropped from your vocabulary.

I've had to hold down our daughter at 9 years old for vaccinations. It can be done with a toddler. Mention the situation when you arrive and ask for someone who's experienced at working with difficult children. They're pros at this and I promise you a fussy toddler is in their wheelhouse. They'll know how to do the blood draw in the best possible way.

I'm really surprised that your pediatrician doesn't have experienced staff or that you haven't been referred to a pediatric clinic. This is so common. Please go back to them (by phone is fine) and talk to them about the problem. If you don't feel this office is competent to deal with your child, now is the time to get another pediatrician. If you think you are a bit anxious about this and maybe your child is picking up on it, you might have someone else come with you to hold the child. I've never heard of kids who can't get shots or have blood drawn because they are too strong.

I'd also suggest that you edit your question so that it just goes to towns in your area. NJ is a huge state, and if you're near Trenton, it doesn't help to refer you to someplace in Ft. Lee. But Trenton is a big place, and you're within driving distance of major children's hospitals in Philadelphia, so you should have plenty of options. But I wouldn't think that this would require you to go to those lengths.

Staff are trained to restrain kids. A 15 month old isn't hard. You wrap them in a blanket, like a burrito, with one hand/arm sticking out. Then one person holds the arm still while another sticks and draws the blood. It's not a big deal. The kiddo needs the blood work done or the doc wouldn't have ordered it.

It's harder on the moms than the kiddo...

Our children's hospital has a pediatric clinic just for kids who need lab tests. They are totally trained for pediatric blood draws.

If your child is being treated or seen at a pediatric center, then the phlebotomists are going to have a lot of experience with kids and doing blood draws. If it is just a general clinic, then you could ask if someone has more experience than another with children.

As an aside, just because your child is "strong willed" doesn't mean that insurance will pay to have a specialty clinic do the blood draw. Also, if this is something that needs to be done with any frequency, then you will need to address how to handle it going forward. Personally, I've had kids with blood draws from an early age (a couple of them had quite a few) and really never had a problem with it. It sucks, but the bigger deal you make out of it now, the bigger deal your child will make out of it going forward.

Good luck.