What do I do with the extra "ground chassis" tab on a DC power jack?
I don't think you are correct about a "chassis ground".

Every barrel-connector jack like the image you provide has three pins - Tip, sleeve, and connector-present.

Basically, there is an internal switch that is opened by the presence of a plug in the connector. It's often used to disconnect internal batteries when the device is powered off of an external supply.



Short of a few rather unusual laptop power supplies, every barrel connector I have ever dealt with has been two-conductor only.