I really want to second a dermatologist, especially if an allergist didn't look into skin-related allergies.
Because your options are, and this is from a novice soaper and someone NOT in the medical field, becoming swiftly limiting with soap. You've eliminated, basically, all the options for "cleansing" oils (coconut, and for safety PKO & babassu because of the close relation and that an allergy to one increases probability of an allergy to all). You've eliminated more or less all of the non-cleansing hard oils: lard, cocoa butter, shea butter, and by extension for safety mango seed & other butters (most share a number of proteins and any of those could be the trigger). And you've eliminated high oleic oils.
You've pointed out a couple mid oleic/linoleic oils that work for you... but the problem is if you look into general recommendations, "higher" linoleic/linolenic content in a soap increases not only probability of rancidity, but my understanding is it can accelerate the timeline for rancidity.
And most soapers who do this professionally won't be willing to touch making a product for you for a few reasons:
1) Making a "higher" linoleic product decreases the shelf life, increases rancidity, and means the product isn't one they can stand by ethically.
2) Making a "higher" linoleic product can have repercussions for their product liability insurance if you (or someone else) were to submit a claim if, or rather when, the soap went rancid.
3) With all of your allergies, making a product for you puts them at risk of liability if it triggers
anything.
All of that places the burden of making a product squarely on your shoulders, which I'm sure is no surprise. You ARE
here after all...
But with the increased risk and timeline for rancidity that a high to mid linoleic soap will carry... realistically I see that you'd be needing to make a SINGLE (maybe two) bar(s) of soap once every week to three weeks (depending on size, how long it can cure, shower performance, etc) and then IMMEDIATELY picking up the new bar as soon as cure is done. Trying to make in more bulk than that means you run the real risk of rancid soap / DOS before you get to it, and that means wastage. Lack of hard oils means that the bars sit in molds longer, and may mean longer cure times (a true castille soap requires 6 months plus and some people on here say 12 months plus). So it's also possible that you can't even realistically GET to the soap before it starts to go rancid. Antioxidants like vitamin E or ROE help but all they do is slow/delay oxidation which the source, at least as I understand it, of rancidity.
So... realistically, I think you need to work with a dermatologist and/or other medical professionals to address the issues medically. You might not have an allergy to oleic oils, for example. It could be a different adverse reaction from an underlying condition that needs to be treated, as suggested by
@Quanta.
