Personally I like the idea of a proper backing plate for the chuck rather than using the dog plate, and for a number of reasons. Mounting that four jaw in the three jaw looks a little shaky to me, but I understand it gets done all the time, and as far as we know, gets done without incident. It probably works fine as long as one does not try to work large parts or take heavy cuts in the four jaw. I know that money is tight for you now, Dan, but if I were you I would be patient and obtain a proper backing plate once you can afford to. Here is why:
1. Your dog drive plate is probably a bit thin to begin with. You will notice that its bulk is in the outer rim and around the nose mount area. From the photos it is difficult to tell how thick the face of the plate really is. The dog drive plates that I have are thin - too thin to consider for use as chuck backing plates.
2. To make a backing plate from your face plate you would need to first mount the plate on the lathe, then take a cleanup pass or two across the face to true it up, making the plate a bit thinner yet.
3. Next you would need to turn a shoulder on the outside of the plate, leaving a spud sticking out that would fit closely into the recess on the back of the chuck. Axial forces in play when the chuck is in use get transferred to the mount through that shoulder, not the mounting bolts this way, not to mention that the shoulder centers the chuck on the mounting plate. The shoulder should be as deep as practically possible.
4. Lastly you would transfer and drill your mounting holes in the mounting plate. Depending on how much material was removed, and where, the cross section of material holding those mounting bolts might be a bit thin, making your faceplate mount more dangerous than mounting the four jaw in the three jaw.
I have an eight inch four jaw for my lathe. It's backing plate is machined steel instead of cast iron, and is a full 1/2" thick at the outer edge where the bolts are. It's outer diameter is the same as the recess in the back of the chuck, so there is no shoulder on it. I had to remove the backing plate from the chuck just to see how thick it really is. So now I can clean the entire chuck while I have it apart. Look what you got me into! :rofl:
One other thing to note is that your dog drive plate is cast iron, not steel, as evidenced by the rough slots for the tail of the drive dog. To obtain the same strength as steel, the cast iron would need to be a little thicker in cross section than a steel plate would. A failure of that cast iron plate would be sudden and total - it would not bend if it were unable to handle the stresses. It would simply crack and break, hurling your chuck in whatever direction was handy at the moment. Not a scenario I would want to see or hear about. Most chuck back plates are made from cast iron, but of substantial thickness and heft.
A dog drive plate for your lathe would probably be more difficult to find, should you need one, than a chuck backing plate for the LOO mount. At this point in time you do not see a need for a dog drive plate, but then you don't know what projects might come down the pike, either. I do not often use drive dogs, but I do use them from time to time, at least a couple of times per year, especially if I need something very accurate. I will not let them disappear as they are too handy to have around!
CDCO has new chuck back plates for the LOO nose. 6" for $78, 8" for $94, plus shipping of course. Having some patience and getting the right thing might be very worth while.
http://www.cdcotools.com/ (Not affiliated with them, just a happy customer).
Good luck with what ever route you choose to take, Dan, but please think carefully as you proceed. Time for me to go clean that chuck.