I would absolutely (and do) bring my own, however, yours does sound a bit cumbersome so I would factor that in.
The main thing you need to consider is that in the parks kids that age have a lot of stamina. They are excited and eager and really there is minimal long term walking. It's not like she'll be walking 2 miles straight in any park. She will walk a little and then wait a lot (in line).
The REAL walking issues come at the end of the day, at the resorts, at Downtown Disney, etc...i.e. place you won't have the option to rent. Also it is VERY nice to have that stroller to buzz through the airport rather than moving along at the pace of a toddler.
We bought a Maclearan for our 1st trip with our son in 2006 (he was then 8 months) and it was the best purchase we've made baby wise. He still fits in it comfortably and will for awhile ((he's 2.5, 37 inches and 35 pounds). It is really lightweight, folds with one hand and has a great shoulder strap to make it easy to carry. In my opinion it was very worth the added cost vs. a cheap umbrella stroller because DH and I are 5'8" and 6' and the little ones are very awkward to push and uncomfortable for the kids.
I know there are some companies that rent baby equiptment (baby's best friend?) and they might have a stroller option, but I really do like having it at the airport.