Is there a good way to find out [a battery's] real remaining capacity except draining it by a known current and measuring time?
While I'm curious to this myself and would love to see other answers, my guess is that applying a known load and seeing how it responds is the most reliable way.
In fact a "battery load tester" is a fairly common tool used by auto mechanics when evaluating the health of a battery. You may be able to borrow a simple one from an auto parts store. It's essentially just a dummy load and meter and you'll still have to extrapolate from the voltage under load to an estimate of health/capacity.
Your parts store or auto mechanic may also have a fancier "battery analyzer" or "intelligent battery tester" that does an actual discharge test for you and can display a report when it's done.