Bamboo may surprise you; inexpensive laminated bamboo chopping blocks are available in any home goods store (including the eponymous “Home Goods”).
Don’t put the cart (source) before the horse (sensitivity). The cheapest experiment I can think of: go to a medical supply store and buy a single pack of tongue depressors. After you’ve proved your tolerance, then you can look for sources (what lumberyards are near you — ones that don’t spell “wood” as “fir/pine”). Don’t confuse northern white birch (Betula pubescens) with other birches, they’re lots of distinct species with different properties (and different chromosome numbers).