I am not really sure whether you can run a hand pump *and* your electric well pump on the same well at the same time... talk to a company that services wells and well pumps.
If you can, I know there are hand pumps that claim to be able to raise water 20 feet but I'm not sure you can go any lower than that, so it depends what the water level is in your well. (Frankly if your well is that shallow I hope you are doing regular and thorough water testing before drinking, washing or cooking with it in the house
)
I am not at all clear on what's your concern about keeping the coop floor dry??? Being near a well usually ensures *dryer* ground (just because wells are, or anyhow should be, built on somewhat higher rather than somewhat lower pieces of ground, so water shouldn't puddle there).
I'd really suggest not putting more than just a couple chickens very close to a shallow well, though (that is, shallow enough you can hand pump from it) because you do not want nitrates and fecal bacteria from their poo getting into your wellwater (by leaching thru the soil, moving a short horizontal distance to the well, and then migrating down along the well casing such as it is into the aquifer). This would not be much of a concern with a deep drilled well with a good top and casing but a shallow well it *would* be a concern. Not only do you not want to mess up your own well, you don't want to mess up others' nearby.
Good luck,
Pat