You're right. I did have to remove one of the pads, but then I got the arms close enough together to let the bolt slip out.
Unfortunately, I took a lot of measurements last night, and my memory was faulty. The original bolt head is 15mm wide, not 10mm wide. I inserted the 20mm long bolt and mounted it with a 5mm spacer and a 2mm serrated brake washer, and torqued it down perfectly using the recessed nut. I stopped the bolt from spinning by wedging a 2 or 2.5mm allen key on one side, between the 10mm bolt head and the recess in the plate.
I think it's actually fine like that, but my friend who started building frames in the past couple of years told me tonight that he just got a die set, so I may have him cut new threads and cut the length of the original bolt to make it perfect.