Update: My flight arrived a little early, and a 7:50pm arrival left me with enough time to catch the 8:15pm bus (found out about it when I saw a sign at arrivals, it seems to run on the :15 and :45 of every hour) that goes directly from the international terminal to Hakata Station (EDIT: without any stops in between, even), and in doing so, also made it on to an earlier Shinkansen.