The first thing I learned doing research is that I need some sort of bookbinding press. Cast iron nipping presses are $2000, very big and very heavy, so I wanted to avoid going that route (although I am ashamed to say initially, I considered it).
Instead, we're going to try making our own makeshift one using two cutting boards ($10 each from Amazon) and a bunch of hardware that cost about $6 total from Lowe's.
The idea is that we drill four holes in the cutting boards, put the screws in each hole with washers and wing nuts for the clamping action.
First, I measured where the holes were going to go. I went with about 15mm from each corner. I don't want to get too close to the edge and have the corners of the cutting boards crack. Then I put masking tape over where the holes will be made to further prevent cracking, I think? I don't know if that's really a thing.