How should I attach the grounding conductor for a replacement switch?
You would be best advised to shut off all live power feed to this box. Then pull the existing bare copper GND wires out of the back of the box and add an additional bare copper wire pigtail to the bunch. Then reattach to the box with the proper green grounding screw.

The new pigtail will get wire nutted to the green wire on your new dimmer switch using an appropriate green twist on wire nut.