Clean Looking Patches

I am asking this for my embroidery person, so hopefully I don't come off sounding dumb.

We have a client that wants us to do embroidery on light weight (3.5-4.5 ounce) poly polos with a logo that has 14K stitches- 6-7 colors. We can reduce the puckering to an acceptable level we embroidery it. Our concern is how the puckering is going to be after they start washing it, we feel it will start looking like crap before long.

My thought is find a clean, thin embroidered patch and sew it on. I want a patch that doesn't look like one on uniform supplied shirts. These polos are for a hospital staff and they need to look sharp.

Any suggestions on a vendor that can do a patch with that many stitches and still look almost like it is embroidered on the shirt.

Hopefully I have explained this to where it makes sense.

Thanks
Larry