I would not fertilize in small cups. I also would not trust the big box "feeds for months" stuff, either.

In my experience beans - Blue Lake Bush is what I grow - do best sowing directly into their final growing spot. I also do container gardening. I would get those sprouts to their final growing spot and let them adjust and feed off the big box mix at first, then add some inexpensive 10-10-10 or 13-13-13 - easy to find. Don't add high nitrogen or you'll get too many leaves and too few beans.

All said, there's always the "it depends" qualifier, as growing involves light, soil structure, water, temperature, fertilizer, organic life in the soil, aeration... I'm sure I'm forgetting a dozen or more things offhand.