Is there any way to use lathe dogs with a regular three jaw chuck or do you need the slotted plate as shown in your photo? I seem to recall seeing a photo of a lathe dog being used in a chuck, but maybe the plate was mounted in the chuck. Do you have any more photos? I'm quite curious about this.
Yup....
First, center drill your stock as your normally would. (Don't forget to do this first!!!)
Now, mount a different piece of round stock in your three jaw chuck.
Turn a 60º point on the end with your compound. (This now becomes your "center". DO NOT remove the center from the chuck!!!)
Place the lathe dog on the stock to be turned.
Mount the stock between your new center and the center in your tailstock.
Brace one of the jaws on the 3-jaw chuck against the leg of the lathe dog.
If you remove your new center from the 3-jaw chuck, you will have to re-turn the 60º point on the end to ensure concentricity.
Andrew