I pretty much agree that this is not going to be easy. The best international airport to connect by train to Hiroshima is KIX but AA does not fly there and their partner JAL is going to make you do that nasty NRT/HND link. By far the best connections from the west coast to KIX are via UA, SFO-KIX, but from your profile I suspect you would rather walk.

Seriously, if I were doing this (and I freely admit to my UA-dom) I would connect SFO-KIX and then take the airport deluxe train (the Haruka) to ShinOsaka and connect to the shinkansen (bullet train) to Hiroshima Station. Unlike the airport, Hiroshima train station is right downtown.

Barring a trip on the blue side, you might check AA partner JL for connections to KIX, ITM or even NGO, although the latter will involve over 3 hours of trains to Hiroshima. Or, just suck it up, spend the first night in Tokyo and make the HND-HIJ connection in the morning. That still leaves you with the long cab or bus ride into town.

Once you have an idea of how you are going to get to Japan, we can probably help with the most efficient train connections. We do that a lot.