Here are the ingredients I found --
Arachidyl propionate, camphor, carnauba wax, cetyl alcohol, fragrance, isopropyl lanolate, isopropyl myristate, lanolin, light mineral oil, menthol, methylparaben, octyldodecanol, paraffin, phenyl trimethicone, Propylparaben, red 6 lake, saccharin, white wax.
Paraffin is well below "fragrance" in the list, so I imagine it's only a tiny % of the whole.
Would it make sense to do a homemade type lip balm using carnauba wax and/or paraffin with light mineral oil (and/or veg oils of choice)? Maybe a small touch of lanolin?
Since your mom is super sensitive, I'd think about using Lanisoh lanolin (found in the baby supplies section) unless you know she's okay with regular lanolin. Lanisoh is super refined to be non-allergenic. It's the impurities in lanolin that people are sensitive to.
I make a lip balm I'm pretty happy with. It doesn't use the usual butters you see in most. I started with roughly 1/3 wax and 2/3 liquid fats and tweaked the proportions from there to get the right consistency -- glides on the lips, stays put for a reasonable while, and isn't melty in the pocket. The final proportions I use are about 68% liquid fat (sunflower and jojoba), 1% essential oils, and the balance being beeswax.
The camphor and mint in many of the mainstream lip balms make my lips tingle and even burn a little bit. I hate that feeling.