Given the requirements you describe, I'd say your best bet would be to use a Kiddyboard, but quite honestly, that's more an option for a 5-6 yo than a just turned 3 yo. I think that a 3 yo is likely to need a stroller pretty much f/t at WDW.
I think that renting a double offsite from a company like ABBF is a good option, as others have already suggested. (Depending on where you are staying, getting 2 sleeping small ones back to your resort could be exhausting without a stroller spot for each of them.)
Just curious, why the prejudice against umbrella-fold strollers? If you are hauling a big tank around for the baby, an umbrella-fold that you can hook between the basket and handle of the bigger stroller when folded is the most logical type of extra stroller to take with you. (Well, that or a Combi, but you said you wanted inexpensive.)
My recommendation for your cheaper second stroller would be a Chicco Caddy, but that is an umbrella-fold, and it's not so cheap that it would be disposable; it will cost around $50. The bright side is that when your younger child is 3, you wouldn't have to haul that big tank; you could take your lightweight Caddy, instead (and perhaps put a buggy board on it for the 5 yo!) They do have decent resale value; people always want a larger reclining umbrella stroller w/ a canopy.
http://www.amazon.com/exec/obidos/tg/detail/-/B00005QXQC/002-9077549-3558404?v=glance
(There's a $10 off sale on strollers at
Amazon right now, not a bad deal.)