I doubt it's anything to do with the timer.
Doesn't seem to be anything inside the pump, the flow through the pump is great.
Did you remove the pump hoses and look inside the pump?
It's hard to tell from the video, but it almost sounds like the idler arm is tapping on something. It's been awhile but it seems I had one like this years ago that took a bit to figure out and the noise ended up being the idler arm tapping or hitting on something it shouldn't be making the noise, maybe from a stretched out belt or something, just don't recall off hand at the moment.
How's the belt look?