I have the same problem from time to time. Any menthol burns me and recently had a problem with itchy jaw line. I believe that was caused by Tabac aftershave. I don’t have a particularly sensitive skin but some things set it off. I use mildly scented soaps and creams now. You might try Palmolive classic shave cream. It is a great performer, inexpensive and mildly but pleasantly scented. It is made in the EU where many chemicals are banned that are still used in the US.
I get the same exact reaction with Haslinger Scafmilch shaving soap. It only burns on my cheek bones. It's just one of those things.
I think it was because I wasn't use to a brush and soap. I've been using it ever since I posted this and it's just about stopped bothering me. If I lather up higher than I usually shave, my cheeks turn a little red and get a little burning sensation.