My friend/colleague who is an object conservator uses Acryloid B-72 for similar conditions. It’s non-yellowing and can be reversed in acetone. Hold a brush dampened with diluted B-72 next to the silver edge and the adhesive will be wicked under the band via capillary action. You might put some painters tape on the coral adjacent to the silver first to protect it. It can be purchased at Conservation Resources: http://www.conservationresources.com/Main/section_34/section34_16.html

While I was waiting on the above info, I looked at the details you posted and noticed some of the teeth (or zig zag detail) appear to bend inward; perhaps this was done to hold the bands in place. The third image shows the teeth or zig zag detail bent a little toward the coral. One could tap all of the ends of the teeth on the loose bands ever so lightly with a small hammer (with the head covered with a thin fabric) to bend them slightly inward toward the coral. In doing so any concern about adhesives on the porous coral and the paper loosening and falling out is eliminated.

I read Jake's post about pitch and carbon being the original adhesive placed under the bands before they were brazed in place. How does one get it under silver and gold bands that are already in place? Is it thinned so it could be applied as a dilute liquid as described above?