I personally think renting the double stroller ahead of time may be more of a pain in the neck than anything. You are going to have to fold and unfold it every time you get on the bus, monorail, etc. and with no one to help you, you will not have enough hands to do that and hold onto your twins.
Might it be to your advantage to just rent the Disney double stroller at the entrance to the parks, thus giving you two free hands to get the twins on and off the bus or monorail? Although the Disney strollers may not be as comfortable as your own, at age 4 it's not like they are babies. They will probably spend a good 50% of the time out of the stroller anyway and comfort may not be your main concern.
My best suggestion is...can you bring someone else with you? Even a teenage niece or nephew? Your biggest need is just another pair of hands.
Or, you might want to switch to a monorail hotel (GF, Contemporary or Polynesian). Being on the monorail line might be more convenient for you in the long run.
Perhaps you might want to dial the trip down and only do one park, or maybe just the Animal Kingdom...or even spending the day at Downtown Disney is great fun, and the hotels are an experience on their own. Is this a two day trip or an eight day trip? That matters. The longer it is, the more difficulty you are going to have.
And I once did the Magic Kingdom on a cold February morning with a 5 yr old, 4 yr old and 8 month old all by myself, with a double stroller brought from home. Wasn't fun, but I did it.
Good luck and have a great time!!