Just a quick note,
I looked up the MSDS or Materials Safety Data Sheet for Tegaderm dressings. Most tape adhesive is acrylic of some variety. Tegaderm's MSDS describes a trade secret form of acrylic, I assume is their adhesive. The reason I'm not exactly certain is they don't say which material is doing what function, but a sheet of acrylic in a bandage sounds odd but not impossible. They list a silicone polymer that like acrylic could be sheet or adhesive. Silicone can breathe in certain forms.
They do note that a mild reddening of the skin is not uncommon. It seems that is what most people complain about. Acrylic plastic, especially tape adhesive is a common culprit in the allergy list. Silicone's are inert in most cases. If the average person were to pop a piece of acrylic plastic in their mouth, say the size of a dime, it will begin to irritate the corners of the mouth and other places where we have a little tender or cracked skin or mucosa is exposed. It tastes bad, it burns a little and irritates easily. This is Plexiglas, the magic plastic of (older) jet fighter canopy's that regular bullets bounce off of!
Since the rash is appearing all over, wherever a bandage is used, I'd see if they can use products without acrylic adhesive. Silicone adhesives are better but also not 100% perfect. Unfortunately, we can develop an allergy to nearly anything. The other products used in cleaning the wound, the tubing itself in the PICC line, and so forth.
I wonder if an allergy specialist might bring a faster and more definite answer to exactly which component(s) are causing problems. We associate allergies with breathing allergies but a systemic allergy should fall into the same basket. Certainly they know how to isolate the culprit quickly and without doubt as to what is causing a reaction.
Best luck
LL