How to handle a glitch gracefully with client

Question

Any professional designers here, I could use some perspectives on how to handle a situation.
Briefly: first time client, somewhat fussy but well-intentioned and nice to work with. I designed an annual report for them, process color, nice piece. The cover image is a color satellite image from NASA, and on the proof some slight horizontal banding appeared in one area that turned out to be a glitch in the printer's proofing device.
I did some very slight color adjustments on the image and on the second proof some diagonal banding appeared in the same area. After discussing this with the prepress chief, I concluded again that this must be another proofing device issue. I did not see this banding on my monitor, so concluded it wasn't in the file.
At the press check yesterday I was surprised to see the banding. I discussed it with the prepress chief again, he said he had noted it on screen the previous day and "thought that T---- was going to discuss it with you." To my knowledge nobody had contacted me (T---- was off that day).
I have never stopped a print run before and was reluctant to do so now (everything else looked A1). This banding in one area of the image didn't seem catastrophic, just annoying to my fussy designer eye. A lot of people might not even notice it. Apparently, it's part of the original composite digital image NASA created.
So the question is, how do I handle this with my client? When showing them the proof with the banding, I said "don't worry, this won't be on press."
My professional sense is that I offer to correct the cover in Photoshop at no charge and have it reprinted (nothing has been trimmed or bound together yet--I've asked the printer to hold off on that). Do I wait and see if my client even notices? Do I point it out first thing?
And with the printer, do I insist they reprint the cover gratis? Or do I share responsibility?
If I had it to do over again I would've insisted on a perfect proof...I was just trying to keep things moving along and I trust this printer who I've worked with before (they do good work).
Thanks for your perspectives.

Answer 1

Thanks guys. I had a good chat with my printer rep. today, nobody pointed any fingers, they basically said they are willing to do whatever is necessary to make things right, except they indicated that they would charge for any additional prepress time. That seemed fair--I can do the Photoshop work and all they need to do is run a new proof and burn a single plate. I would cover that cost.
Later in the day I met with my client and told them everything went fine except for one glitch--I asked if I could wait until after they'd reviewed the proof to reveal it, just to see if it was obvious. They good-naturedly went along (we have a good rapport) and could not spot the problem. When I pointed it out and explained what happened, I said I would be more than happy to fix it, no charge to them, just a little extra time delay in getting the final product.
The person I work directly with felt it wasn't worth fussing over, but wanted to show it to the boss who wasn't in today. So I'll get a final answer tomorrow (he's the fussy one, so I'd say there's a pretty good chance he'll want a reprint).
I feel it has been handled well by everyone to this point. Good learning experience for me.
Thanks again for your input. Any further advice or comments welcome.