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I'd move the coop, frankly
Honest. That way I could have the option of electricity there too. (250' is a really problematic distance for stringing extension cords without asking for trouble...)
But, failing that, if the coop had to stay where it is and be forever unelectrified unless you want to trench in a proper circuit, I would do one of the following two things.
If it were like a dozen or fewer chickens, I'd just carry water from the house every coupla days in lidded square cat-litter type buckets. The lids snap on tightly enough you can use a wheelbarrow if you don't want to manually carry the buckets that far.
Or if it's more chickens than that, or you have physical or topographical issues that make carrying water seem impossible for some of the year, then slice the turf and lay a hose (preferably one of the super heavy duty lifetime guarantee 3/4" hoses, but normal hose will work too if it's decent diameter, none of this 3/8" stuff) in that shallow trench, going all the way to the coop or as far as your hose budget allows. This requires a little work slicing the turf and tucking the hose in as you go along, but is way easier than trenching in a real waterline below frost depth. The hose cannot be used during weather cold enough to put frost into the ground, and if it's a normal hose it needs to be disconnected from the house spigot *before* such weather arrives; but for at least *most* of the year, that'll give you water in the coop at not too awful much work or cost, at least compared to putting in real water line.
Good luck, have fun,
Pat