What if you made something shaped like this
Not sure what you would make it out of, maybe use something like petri dishes and some acrylic tube (the tube would be curved at the top, not square like in the pic). Keep the tubing as short and thin as possible to limit the volume of air. Imagine a more streamlined shape for a finished product. The white square inside the solution would be something that would take up some of the space to keep the solution volume low, but let you keep a large surface area on top of it. You would view it from the angle the red arrow is pointing so that you still had a decent size viewing area with a white background. The shallow squared off shape instead of a funnel should also give less volume of air inside.
Just an idea that I had trying to think of something that fit the criteria of high surface areas and low volumes.