ground wire

Do I need to ground the ashtray bridge if my pickup is grounded from the copper plate to the pickup already? Don't the screws from the pickup help ground the bridge?

In theory, yeah, the bridge plate should get grounded through the mounting screws and the pickup's baseplate. In practice... you don't always get a solid connection It doesn't take much corrosion to make that ground flakey.

An eyelet pressed up against the bridge with the pickup's mounting spring is a better solution, and builds in some redundancy.