When you make that decision to purchase polo shirts with a logo and the fabric is thin, you may wish to pay the extra shipping expense and sew ONE shirt to see if it works for you. If you are not satisfied with the result, you could thin out your logo, removing some of the detail so that the logo is lighter. For example, the FORD logo is an oval with the word Ford inside that oval. They will embroider inside the oval for jackets and leave the inside of the oval alone, exposing the polo shirt's fabric inside the oval and only sewing the one word inside,
I am not a big fan of the tear away or cutaway fabric needed to support a logo on the inside of a garment, so a thinner logo may be more attractive.
The embroidery store at True to Size Apparel offers setup / digitizing at less than $8 for a 5,000 stitch logo, while your polo shirts with logo order can cost as little as $2.99 to embroider each shirt. How does that make your budget feel? Nice!