Hi. I don't know if this is relevant to your problem but I'll tell you how we move water around. Our holding is on a gentle slope with the house near the bottom end and a long shed at the top. We use this shed roof to collect rainwater in 2 x1,000 litre barrels. The water then goes by gravity from one barrel to all our sheep drinkers, which have ball cocks, and from the other barrel to a hosepipe which waters my veg garden, which is about halfway down the (gentle) slope. Further down, near the house is a long barn, and again we collect the water from this roof in 2 x 1000 litre barrels, which are linked by a hose pipe (so the far away one fills the nearer one once it has been emptied). The nearer barrel in turn fills a 40 litre drum (again connected by a short hose pipe - it has to be 'primed' by running water through with the hose lower, then a thumb stuck over the end and lifted back into the barrel) - this has a small water pump attached which pumps water back up to just above my polytunnel (it takes maybe 5 mins or so to pump up each barrelful) , where it is used to fill yet another 1000 litre barrel, set up on a plinth to give it a head. This one is used to water the tunnel, via an ordinary hose as well as seep hose (not very successfully as the plinth isn't high enough).
I am not sure from your description whether you need a continuously running pump, or whether you could do as we do and collect your water in a larger container then pump it up whenever you need it. We used to carry water up in buckets with lids on, very briefly - it's unsustainable so I invented the system we now use. If you buried a barrel near the trough, or as you say pipe it downhill first, you could pump it up whenever it was full, or whenever your container at the garden was emtying.
So far collected rainwater using this system has been enough to water all the animals and plants. We have a well for our own use but it has run dry in the past, so we would only use that water for the animals in a real emergency (and in the worst of the winter) The 1000 l barrels are I think now about £160 from the Tank Exchange but worth the initial investment.