So I've added the first layer, which worked out to about 5" all told. I also went heavier on the mulch than soil, not for any good reason, but because I needed all the soil for another project. I don't think the taters will complain all that much.
At 5" every two weeks or so, by mid-June we should reach the top of the barrel. HR, bush beans, and marigolds will follow immediately.