Asked OH. In his view the tree is definitely a Willow, but although he thinks it's Crack Willow, he's not 100% sure. The problem is that Willows are known to hybridise quite extensively and so he said he would need to see the tree himself, to confirm it's habit, before being able to say definitely Crack Willow. However, for your purposes Willow is fine
He said the other tree in the foreground of the 1st pic and to the right is not (as I thought) Field Maple, but Sycamore, which is a Maple and is fine for rabbits.
https://www.thesprucepets.com/safe-woods-and-plants-for-rabbits-1239351
What a lot of new forage your rabbits can now have