The only things I remove from a bale of hay are seeding docks and thistle. The docks are not eaten and won't compost or get eaten, so will regrow from the seeds. The thistles just have tiny very sharp bits that cause all sorts of issues when they stick in skin, eyes, my fingers...
I'm not keen on buttercups, either, but they are not toxic once dried. They won't get eaten, though.
That's the joy of getting bales of hay from local farms. Cheaper, fresh, local - but sometimes need a little work. On the other hand, it does mean the fields were not full of herbicides and I would rather have the occasional bit of weeding to do than have 'perfect' hay which is possibly full of chemicals.