Unfortunately, unless you pay to upgrade you will not be able to keep the room. I personally would book the room to be at the resort and ready to start your plans on Mon., without having to stay elsewhere and take ME in the morning. If you had to transfer rooms for your new reservation, I would pack what I needed for the first nt./morning in my carry-on, and when you arrive late nt., you could just ask baggage to hold your luggage till the next day till your new room is ready. In the morning you could bring carry-ones down to be held, and hit the parks. The luggage will be delivered to your new room once it is ready. Also a regular room could become available and you could change to that, and might be able to keep that room when you switch to the new reservation.
Also, you might prefer a preferred room at the All Stars if your budget allows it(not sure if your reservation through Cheap Tickets can be changed). Never felt we needed it at CSR, but stayed at POP in Dec., and felt it was worth it to be close to front for buses, so quick to get out in the morning, and especially at nt. coming back. Also liked being near the food court for refillable mugs, so DD and I could grab a drink to take on the bus in the morning, and could grab one on the way back to the room at nt. If you haven't stayed there before, you could look at a map to see the different buildings, and if you are not booking preferred for the rest of the trip, you could pick a regular room that is closest to what you want(pool, food court, bus) and add that as a room request.