Congratulations!
I've never embroidered on this type of garment, but I would iron some Floriani Dream Weave before to add some body to the garment, then use a cut away no show mesh, then iron more of the foriani product to prevent the little one from being scratched by the embroidery. Let us now how you get on.

Hi
Use Sew Stable Polymesh no show stabilizer. Or any other brand that is a polymesh.
Sew Stable Cutaway Polymesh no show stabilizer, Fusible Polymesh no show stabilizer, tearaway Polymesh no show stabilizer
Its very soft and is made for being next to the body.

I use a nylon tricot interfacing from the regular interfacing area. It has a one way stretch. You iron it on with that stretch going opposite of the garment stretch. I then hoop a tear away & spray it with adhesive & simply lay the garment on it to embroidery. After I remove the tear away, I iron on another layer of the tricot....direction does not matter here. This layer covers any roughness of the embroidery threads. I do have to go back & iron this after a few washings because you can't use a real hot iron to attach it. It tends to loosen at the edges after a few washings. I use this method for all children's wear where the stitching will touch their body. I use the tricot as a base layer for all knits.

I have a 7 year old great niece that swears by this method & knows exactly how to fix bought items that scratch her too! Shortly after I first learned this method, she was trying to tell my daughter the secret way I fixed things & my daughter had no idea what she was talking about. They came to my house for something & she went in my sewing area & got a piece of the interfacing & took it to my daughter saying "This is what you need". We laugh at the 7 year old teaching the 37 year old new ideas.

I use water soluble stuff. wash it once and it's all gone. Plus I always use poly thread not rayon as poly takes bleaching better on kiddy clothing.

A good professional backing for clothing like this is M.E.T. Pro Specialty #90 from Midwest Embroidery; it's a web mesh weave and very strong yet lightweight and soft, (www.midwest-thread.com). If you want a very soft tear away, use Hollingsworth & Vose #1885, we use this on childrenÂs school uniforms and they wear just fine with no extra work. Sometimes you can use an iron-on to cover the stitches as others are suggesting here, however you will almost never see this on ready-made products. If your digitizing is correct, adjust your stitch density a bit low and the embroidery should not be heavy enough to scratch if you have the proper backing. Try to find a close match to the fabric (like at walmart) and test it several different ways till your happy before you run the real thing.

Good Luck