It sounds like either the adhesive is the problem or it is a combination of adhesive and whatever the material in the tape is. Testing is good, but you also need to find some type of tape that won't cause a problem. One of J's friends when we lived in OH was allergic to almost everything. Her mother found that the only tape they could use for bandages was masking tape. They tried all the medical tapes and she reacted, but she got into a roll of masking tape and had no problems. So from then on her mom carried a roll of masking tape everywhere, kept it with gauze pads in the kit with her benadryl and epipen and also put a roll in her medicine kit at school (plastic box with her medications in the office) because putting a bandaid on the little girl gave a bad enough reaction to need the epipen and 911 call. At one point the girl ended up in the hospital for some surgery and her mom insisted on them using the masking tape - surgeon had a fit at first until he was an idiot and tried reg tape and caused a reaction.
At a point when she is otherwise stable, medically, try a small piece of some type of tape on her skin. Take it off right away if a problem happens, but it is important to find some type of tape to be able to use if she ever needs IVs or something.
Does that vet wrap stuff have latex in it, or cause a reaction? I know they sell it for people, it is stretchy and sticks to itself but not to skin or clothes. The lab where my doctor has bloodwork done has switched to using that instead of bandaids because so many people have problems. It is a lot cheaper at the farm and home type stores (like Tractor Supply, etc...) than at the drugstore, and it also comes in a lot of different colors than tan if you get it at the farm and home store. We love it - it is great for support for a sore ankle, etc... if you can't find an ace bandage, and if you can't find clips for the ace bandage you can use a lenght of vet wrap to go all the way around to hold the ace bandage in place.
Sorry she has to deal with all of this!