Came to this question late since we were gone, but seeing your idea about the tape, I wanted to note that tape is going to leave residue inside uniform over time, and that residue could create issues (discoloration, pilling on the fabric) when you wash the uniform (or when you get it dry-cleaned). And your daughter may sweat more where the tape touches her skin, or even get rashes from having the tape itself next to her skin, especially if there is latex or other substances in the tape to which she might be sensitive.
I understand why she can't wear a separate camisole under her uniform top, but I recommend that you either find a fabric to use as a lining and hand-sew it in place, or you get a camisole and basically sew the camisole into the top as a lining so it doesn't show. Cut off thicker portions like the hems around the armpits of the cami. You can cut the stitches and remove the lining when it's time to clean the uniform if you need to.
I have had to do similar things to dance costumes for my teen and have had to improvise linings like this.
One other thought about using tape -- tape will not really flex and move with your daughter's body the way that a soft, flexible fabric will. She might find that the tape would affect her movement, especially if it's wide enough really to cover seams etc. I'd really go with sewing in a simple, inexpensive camisole so it does not show but covers the entire inside of the top. Don't be afraid to cut it as needed, and get several so you have replacement linings ready. I would go with a cotton or mostly-cotton fabric so it breathes a little and doesn't make her sweat too much.
