I've also been diagnosed with EOE and have avoided beer, ovaltine, Ben and Jerry's brownie ice cream, cheese nips, and anything else with malt for the last 20 years for the same reasons you have.
This may sound strange, but if you have the chance, try a small taste of an ale like Rodenbach Grand Cru (or other Flemish red brown ale) to see if your body reacts the same as it does to beer. Where I live it's available by the 750ml bottle at Costco. Yes, it does have malt, but immediately upon trying it (and smelling it even) I could tell I wasn't allergic to it. I think it has something to do with the very long fermentation process used in making it. If it works for you, then there are plenty of sources out there with home brewing this ale.
Good luck and I hope it works for you!