We always let DME handle the bags on our late night arrivals. We arrive VERY late, (land at 11:30, to the resort by 12:30)
We tag our bags, and then bring a carryon with what we need to go to bed and get out the door the next morning.
We have never changed resorts, but in the OP case, here is what I would do.
Tag my bags, have my carryon, then when I checked in tell them we're transferring to SSR the next morning. The next morning, I would take my carryons to Luggage Services, and ask them to transfer my carryons to SSR and remind them that we have ## pieces of DME luggage that also need to be transferred to SSR.
Then I would head for the parks. Later that day, I would check into SSR, and once I got in my room, call for my bags.