It’s possible, but it’s a bit of a hassle. With the mounting plate, its easier to use a screw and nut, which I assume you’re doing because it’s a shoebox.
Anyway, you can still mount through to those two tabs on the left and right in this picture. I don’t know if that 2mm from the mounting plate will drastically effect the height.
I was searching through my stuff today and i found a Shadowblade stick for the PS2. LOL this stick was so horrid, but maybe its possible to mount a JLF onto this thing? I don’t know, ill have to find some time to crack it open