then switch to DVC rental....
[I'm assuming you are renting from an owner or site like David's, not renting directly through Disney...]
Oh that makes it easy! There is no minimum number of days, nor is there a ticket-purchase requirement, when getting the dining plan while using DVC points (either yours or someone else's).
So for your POP stay, you'll buy the tickets you need for your vacation and get the dining plan that makes sense for you (quick service, normal, deluxe) over those 4 nights. Those credits will be good until midnight of your POP checkout day. When you switch to the DVC rental you will have a shiny, new set of dining credits (again, quick service, normal, or deluxe) for the nights of the DVC stay. Just keep on using your ticket (you should be able to put it on your new KTTW if that's important to you) that you purchased with the initial POP stay package.
You'll just have to work out putting the dining plan onto the DVC stay with the owner; they will have to arrange it for you, and they have to pay it in full when they arrange it. So you'll have to pay them, somehow, for that dining plan. The details are between you and the owner.