Welcome to the G.
Using the spring clip on the back is the safest method.
I remodeled a case I previously made for a golf club display for about the same number of tournament money clips by making hanging racks from strips of mat board that were hand covered with a suede cloth fabric. I didn't try to make individual openings in the mat for the display, especially since the client wanted to be able to add more to the display in the future.
Here's a profile diagram of a single rack.
The racks were wrapped individually with the fabric starting on the inside top and down the face, then onto the support board behind the rack below.
These were about 24" long and the case I was converting had room for 10 racks plus end pieces top and bottom. they were spaced about an inch apart vertically so you could work the clip on to the rack.
The drawing isn't proportional and the design will probably differ depending on the size of the clips. These were for 50 years of the same tournament, so the rack size was determined by the largest money clip, as the design changed over the years.
Here's a quick drawing of how the racks were positioned in relation to each other. I also remembered that I had beveled the lower edge of the lath to make a smoother transition from the face of the rack to the back.

Good luck with your project!