I can't see how it isn't "traditional." Having lived in Korea and Japan both, I saw quite a few "traditional" homes and buildings with hard wood floors. Not every room had tatami (in Japan) as they are quite expensive. Certainly too expensive to use as padding for martial training... That's what grass and dirt is for!
We train either on cement, grass and dirt (outside), or on a carpeted concrete floor (no padding - nice and firm for falls, nice carpet burns when you move incorrectly).
But train how you like. If you are continuing at that school, I'd recommend a comfrey tea (2-3 handfuls of fresh, crushed comfrey leaves placed into a pan of water, gradually brought to a boil) applied topically to the offended extremity. It sounds odd, but it works. Other than that, tape the blisters, and keep training...