In the second image, why does the plate disappear and what am I seeing in the space where the plate should be?
The plate "disappears" because it's out of focus. At short focus distances, depth of field is always very shallow. When you focus farther back, the edge of the plate goes out of focus, and because it's so narrow it seems to disappear completely. I approximated your situation using a zip tie held in a spring clamp, and you can see that as the focus moves farther back the zip tie also goes far enough out of focus that it starts to disappear:
I know of three options for increasing the depth of field in your image:
smaller aperture: Moving to a smaller aperture, like f/16, will help to increase the depth of field. At such a close range it won't help as much as you'd probably like, though. Using a depth of field calculator you can see that your 105mm lens at 30cm distance and f/6.4 gives only 1.3mm depth of field. Switching to f/16 increases the depth of field to 3.2mm, and at f/22 you get 4.6mm.
focus stacking: You can take a series of photos with the focus changed slightly from one to the next, and then combine them using focus stacking software to create a final image that's "in focus" over a the total range of all the images. I'm not sure this is a great option for your particular case, though — at the very least you'd have to validate the process to make sure that the stacking process doesn't give you incorrect results.
greater distance: By moving the camera away from the subject, you can increase the depth of field in a single shot. Using the calculator again, you can see that increasing the distance to subject to 100cm gets you almost 20mm DoF at f/6.4 and 49mm DoF at f/16. The down side is that if you want to keep the subject the same size in the final image, you'll have to crop the photo. Using a longer lens won't help — the longer focal length will offset the increased distance. Following is a photo I took of the same zip tie from maybe 6x greater distance, in which you can see the increase in depth of field. But I had to crop the image quite a lot, so there's much less resolution:
(Full disclosure: the image above was taken using a slightly smaller aperture, f/4, than the preceding images, which were taken at f/3.5. The change was inadvertent, and I don't think it changes the larger point, but it's still important to point out.)
I can also think of a few other options that don't involve changing the depth of field:
use your second image: There's a region in your second photo where the interface between the water and the card is in focus, and maybe that's all you really need for your measurement. I'm sure it'd be nice to have the edge of the card in focus, but if it's not actually needed then maybe you already have what you need in the second photo.
photograph the card's shadow: You could set up a collimated light source at the far end of the tank and in line with the card. A collimated source produces parallel light rays, and if those rays are also parallel to the card then they should form a sharp 2-dimensional image. If you project that image onto a semi-transparent screen, you could photograph the shadow of the card and interface.
use a laser: If you hit the interface with a narrow laser beam from a known angle, you could measure the angle of the interface by seeing where the beam is reflected. Move the laser to measure the angle at different points determine the shape of the curve. As a bonus, studies show that experiments that involve laser beams are much cooler than those that don't.