Well, the word comes from the Latin "indurare", to harden oneself (against something). So I'd take this as implying "endure" has a conscious decision behind it. I'd also suggest there is a time period implied ('duration' has the same word-base as 'endure'), so the pain that the dogs do feel or not is stretched over some time. But this is just my own impression. If it's just a matter of feeling any pain or not, than just use "feel".

to endure=to withstand or bear something for some time(usually a long period of time but not always). ex What can't be cured must be endured. another example prior to anaesthetics,solders were given a bullet to bite during an operation to endure pain. in fact the idiom "to bite the bullet" means to endure pain.

I'd use the word 'feel'

endure means to undergo, experience, or survive. Dogs don't experience any pain.