Hello Will,
Can you send the project with the alignment results only (original images are not required) to support@agisoft.com, so that we could check, the distribution of the tie points on the images and any sample original image, so that we could check the shooting settings and image quality?
Background (table cloth) and reflections may cause some problems, but I would also suggest to improve the shooting, by taking more detailed images (from closer distance, while the lens allows to get sharp image) and pointing the camera on the object directly, avoiding taking the images in the following way, where the scene depth changes greatly:
https://londonrevolution.net/Bridge/images/thumbnails/441A0404.jpghttps://londonrevolution.net/Bridge/images/thumbnails/441A0536.jpgFor such shots it is not possible to get sharp details for every part of the image and some areas will be blurred considerably.
The examples of good shooting, according to my understanding, are the following:
https://londonrevolution.net/Bridge/images/thumbnails/441A0584.jpghttps://londonrevolution.net/Bridge/images/thumbnails/441A0682.jpghttps://londonrevolution.net/Bridge/images/thumbnails/441A0792.jpgProviding that you don't have any limitation to the number of images, I suggest to take more detailed images with better overlap, even though only a small part of the object is captured, instead of making a kind of overview images, which wouldn't give detailed and accurate information for processing.
I would recommend to take several rows of vertical (nadir) shooting lines - similar to aerial survey. For these lines also change the camera orientation by 180 degrees on the neighboring lines, then add the images from the sides (taken from 2 or 3 heights, pointing cameras normal to the object surface), if necessary, add some oblique shots for complex areas with many small elements, but prefer detailed (close) shots instead of overview (distant) shooting.
You may need to use a tripod with adjustable height at least for the side shots, thus you can avoid using flash and provide longer shutter speed (especially, if remote control is available).
The diffuse indirect light is preferred, as the flash is causing glares and hard shadows.