OK, where to begin.
Yes lanolin can irritate, but many other things can too. Keep in this quote in mind; "The dose makes the poison". Most soaps are demure enough that users wont notice anything. Looking for the cause of skin ailments it can be easy to hang the reason on the largest percentage of ingredient x in your soap is an easy pitfall.
Lanolin is about as toxic to our body as earwax. That is essentially what it looks like in raw form on the sheep. It is used in such small amounts in soap (as opposed to tallow, clay, etc) that irritable people never notice it existence.
I see you mentioned fragrance oils, but dont rule out essential oils. They can be as irritable as the fragrance oils.
Ingredient lists delineate the parts used to make the soap from most to least. When contemplating soaps, use that as a guide to help make your judgement.
Also keep in mind, this is a personal journey. YOU will have to purchase multiple soaps, and pair these with multiple blades in various razors to drill down to YOUR ultimate combo. Some people can be reactive to an ingredient but never know it, because they need a big dose to know it. Yet others only need a small exposure to a small dose to know that they are highly reactive to a certain ingredient.
Me personally, I am only reactive to cinnamon, and I find that most soapers want to use it in apple scented soaps. Yet, I can use the most mentholated soaps on the market without issue. Be leery of buying Limited Edition soaps as those are made with an excess of ingredient x, and this is what makes them special and limited. Regular run soaps are made with lower quantities of exotics and are more acceptable to sensitive skins. Also, all soapers I am aware of make a plain unscented soap. Most soapers have a set formula and only add the essential and fragrance oils to this base to create their soaps. With this in mind, when you buy one of these bare naked soaps, you are still getting a high quality shave soap without all the other concoctions added to make the other soaps.
Disclaimer
I know that soapers can/will change formulas based on the EO/FO that their recipe calls for. I am aware of my generalization, but I believe it applies for my analogy to help the OP and any others that read this.
