You have to decide if you want to make life easier or spend the least money.
I have the same dilemma with a 4 & 6 yr old. They dont use a stroller EVER at home, but the last few years in WDW we have always used a double stroller and all the kids hop in and out of it all the time. SO if i were you I would either rent or bring a DOUBLE. This way one child isnt whining that they want to get in, and if you need to cross the parks to get a fast pass, you can do so pretty fast if little feet are not walking.
That being said, you have to decide to rent or bring the Double...and what is more important to you. Saving money or making the trip a bit easier.
1-Stress free- renting from WDW..get the multi-day rental (just a bit cheaper than paying per day) Since you are staying offsite, and probably wont need the stroller outside the parks, then you could make it even easier by renting from disney, to avoid travelling at all with it. No buses, no airplanes etc. If you park-hop you just keep your ticket and can pick up another stroller later in the day at the next park.
2 Cheaper option - rent from an Orlando area rental place and you dont have to fly with the stroller but you have it delivered to your hotel and can use it at all times. I think they run around 80-100 for the week?
3-cheapest option - bring the double from home. You can check it as a bag in the airport (avoiding security) or bring it all the way to the gate and check it there. It will allow your kids to be a bit more comfortable (the disney strollers are a bit hard) and if they fall asleep in the parks you can push them to the car at night. - all the same reasons as above.
Also, 2 yrs ago we had long delays on our flight home and the kids wanted to go to sleep in the airport while waiting! It was so great having our stroller that night! We expected to have them home in bed at 8pm and didnt get home until 1am. we were so glad we had the stroller with us - especially while waiting for our luggage to get off the carousel at JFK. Our tired, sleepy kids were asleep in the stroller!
Sorry if this confuses you more! You just need to decide what means more to you - some people would rent at disney for sure just to avoid the hassles. I on the other hand would bring one from home to save more money!