I know that you have made your decision not to do a split stay, but we have done split stays almost every trip with little ones, and have a different perspective than the other posters. The first time was with our 5 month and almost 3 year old sons. We stayed at BLT and BWV in May, and then in September we did another split stay at BLT and BCV when they were 9 months and 3 years old. Having said that, 4 nights in each resort is the minimum we will do. Any shorter, and you don't really feel settled before it is time to move or check out.
For us, we enjoy feeling like we are getting 2 vacations out of one, and by staying at both MK and Epcot resorts it really is the best of both worlds. We only had to take a Disney bus to Animal Kingdom, which was fantastic. We have a Phil and Ted's double stroller (not side by side), but it is not the smallest thing to cart onto the buses. I can't even imagine not having the stroller that my boys are used to, and the most comfortable in while at Disney. We don't mind walking though, so whenever we can walk instead of taking a bus we will.
As far as packing up and getting ready for the second resort - it really wasn't that big of a deal (we didn't have a car with us as we flew). The night before the move, I packed up the suitcases. What I did was pack all of the souvenirs or clothes that we weren't going to need for the second part of our trip into one suitcase so that when we arrived at the second resort there was no need to even open that bag. If you are staying in a 1 bedroom you will have a washer and dryer, and therefore won't have to take as many clothes, blankets, etc. (an absolute life saver when travelling with little ones). The morning of the move, we called Bell Services, and had them pick up our bags. We then headed out for breakfast, and to whatever park we were going to for the morning. Usually around lunch we headed to the second resort to check in. By the time we checked in, and got ourselves oriented the rooms were ready.
I am sure whatever you decide, you will have a fantastic trip with your little one! Just remember, now that you are
DVC members you have many, many years to experience all of the resorts!
As far as your concern for feeding - the Baby Care Centres have private nursing rooms with rocking chairs. It gives you and your baby privacy and some quiet time. The Baby Care Centres are really great - we always make sure to head to them when it is time for a fresh diaper. They are always clean, and less crowded than the public washrooms.