I looked up the ingredients in the balm you loved and it mostly contains very moisturising oils and butters like shea butter.
A brand that behaves very well with my sensitive skin is Avene, they are known for having skincare with few ingredients and a whole line specifically for sensitive and allergic skin.
They have a "Tolerance Contol Balm" that contains shea and squalane, an oil that closely mirrors the skins natural oils.
It's here: https://www.eau-thermale-avene.dk/p/tolerance-control-balm-3282770138856-c1428227
Make sure you get the balm and not the cream as they are completely different formulas.
I would also consider trying the cleansing lotion they recommend, as it would be very gentle on the skin. It is possible for some people to react to water, and something like a cleansing lotion that you rub in gently then wipe off with a damp cloth might help!
https://www.eau-thermale-avene.dk/p/tolerance-lotion-3282770142280-c1428227
I have been using The Ordinary's Squalane Cleanser for ages now, and it works the same way. No soap, just rubbing a gentle cream on and wiping off with a damp cloth.