Put some thermal paste on the CPU cooler, where the processor would normally sit. Then mount the Peltier module onto it, so that most of the area is covered.

Cut the copper plate into a size of around 10 x 10 centimeters. Then drill four holes into the copper that match the location for the mounting screws of the CPU cooler mount.

Put some PE insulation tape on the bottom of the copper plate around the location where the peltier element will sit.

Cut out a small section slightly larger than the peltier in the acrylic sheet, then also drill four holes in the acrylic that match the location of the mounting holes on the copper plate.

Put some more PE tape on the bottom side of the acrylic sheet around the peltier cutout and the screws to really make sure everything's well insulated.

Screw the arylic sheet and copper plate together by using some plastic or acrylic screws and standoffs. You can screw it right onto the mounting brackets. I used screws on the copper through to a standoff on the underside and then some nuts and washers on the mounting brackets.

Put some thermal compound on the top side of the peltier module and put on the copper/acrylic plate assembly. Make sure the mounting brackets line up with the cooler's mounting holes of course. The whole assembly should slot right around the peltier module. Then tighten the mounting screws on the cooler's base until the copper plate sits flush on the peltier module. It should not move at all and should have some good mounting pressure.

Solder some 12V PSU cables to the peltier to power them. Extra points for using extension cables, cutting off the male end and soldering to these cables. Then you can always use the female ends of the cables to connect to or disconnect from the power supply.

On the top side of the acrylic plate right around the copper plate, you can put some more PE tape to flatten the edges quite a bit and level everything out on that side. The tape is also flexible and can compressed, so it's ideally also a good surface to put on the bell jar.

Don't forget to mount  the CPU fans to the cooler. You can connect them directly to the PSU 5V or 12V rails, depending on wheter you like it silent or prefer the extra performance. You can also solder in a slider switch to switch between the two "modes".