I don't know of a real workaround, but for the Resort Shirakami (or any rapid train rather than limited express), it's probably easiest to just buy and pay for the seat reservation, 330 or 530 yen depending on the date. Just check the "Purchase tickets such as the Super (Limited) Express Train ticket only" box.

I've done it for the Resort Shirakami and High Rail 1375 (plus others in West Japan) and imo it's worth it, especially around any holiday when those two and some others sell out way in advance.