Black underwear?! Strapless bras?! The dance team doesn't know what it's talking about there. No dance studio or team should ask kids to get things that are not realistically available or wearable for children who are built like ... children. As for dyeing underpants -- you'll need cotton underpants (others won't hold dye), and cotton underwear may be too thick for going under a costume. Plus, you may not get a true black and it may end up too grayish.
Tell the dance team you will get her a flesh-toned leotard -- all one piece. Goes under everything, stays on all the time under all the costumes. Covers the top AND bottom all at once and solves the problem, especially for a child who does not need a bra but does want coverage on top (which is a good idea if the costumes slip!). It's the same as no underwear. I know, because our dance studio has a no-underwear-under-tights rule for performances but all the dancers of all ages use flesh-toned leotards over their tights. It also speeds up changes between costumes because girls aren't trying to be coy and change under their clothes, hide behind someone else etc. -- all the things they do at this age to not be seen naked.
These leotards aren't cheap (look on discountdance.com for some better prices) but are well worth it as your child will use them for longer than she will her regular leotards - these are super-stretchy. Just put her in one and send her and if they say she must have no or black underwear, say, "Next time you need to tell me where to find black underwear for a 10-year-old!"
By the way -- do not get "clear" shoulder straps that are sold with these leotards (unless the dance team specifies them). The clear straps are plastic and actually shine and show up more under stage lights than the flesh-toned straps.