Trenching needn't be costly; I've done quite a lot of runoff-diversion around here (like you, this is a very flat low property smack dab in the path of overland water flow) just with a shovel. In fact the majority of our barn- and backyard-flooding problem was solved by a ditch about 30' long, four shovelblade-widths wide, and 2-3 shovelblades' worth deep. Not done all at once, of course, but you have until next winter to do it, right?
The key is to see if there is somewhere that, if you can keep the water CONSTANTLY FLOWING right from the beginning of the flood/melt event, you can prevent the water from ever backing up. Sure, if you're flooding from your stream overflowing its banks, you are totally 100% s.o.l. and WILL need to raise the coop floor or just build another building elsewhere. BUT, if your water is overland flow headed *towards* the stream, it is quite likely that if you just provide a good channel for it right from the moment the snowmelt starts, you can avoid much or all of your coop flooding. Like, you can run quite a lot of water into a sink and not have it fill up at all if the drain is open; but if you start closing the sink drain even just a *little*, so there is a mismatch between how fast the water runs in and how fast it can run out, the sink will fill up and overflow... even from just a *little* closing of the drain which is easily enough remedied. You know?
(Then you also have to get the snow/ice out of that ditch when a thaw event happens. This can be a nuisance, but, <shrug>)
In terms of the cost of raising the floor, I don't know what aggregate prices are like in your area, but 13 cu yards of roadbase would only be a coupla hundred dollars around here. Plus pavers or something to put on top of it, but you might be able to get something that someone's getting rid of.
Good luck, have fun,
Pat

The key is to see if there is somewhere that, if you can keep the water CONSTANTLY FLOWING right from the beginning of the flood/melt event, you can prevent the water from ever backing up. Sure, if you're flooding from your stream overflowing its banks, you are totally 100% s.o.l. and WILL need to raise the coop floor or just build another building elsewhere. BUT, if your water is overland flow headed *towards* the stream, it is quite likely that if you just provide a good channel for it right from the moment the snowmelt starts, you can avoid much or all of your coop flooding. Like, you can run quite a lot of water into a sink and not have it fill up at all if the drain is open; but if you start closing the sink drain even just a *little*, so there is a mismatch between how fast the water runs in and how fast it can run out, the sink will fill up and overflow... even from just a *little* closing of the drain which is easily enough remedied. You know?
(Then you also have to get the snow/ice out of that ditch when a thaw event happens. This can be a nuisance, but, <shrug>)
In terms of the cost of raising the floor, I don't know what aggregate prices are like in your area, but 13 cu yards of roadbase would only be a coupla hundred dollars around here. Plus pavers or something to put on top of it, but you might be able to get something that someone's getting rid of.
Good luck, have fun,
Pat