You cannot do the dining plan on a split stay unless you book the dining plan for each reservation of your split. In that case you would also be buying one-day tickets for each person in your party for each reservation as well, since that is required to make it a package with the dining.
You might consider booking the dining plan with your first reservation and that way you could upgrade your park tickets for the entire length of your stay. Of course you'd only have the dining plan for the first half of your trip, but if you plan it so you do your more expensive meals on those days, then you might still come out about the same as having the dining plan for the entire stay.