Hi Chris!
I think the umbrella stroller and travel stroller might not be convenient enough to justify bringing along for the trip. There's a good chance your three year old will want to use a stroller since there is so much walking involved. Even the most independent tots want to rest their legs at Walt Disney World! So comfort is key.
There are a few options. Like you said, you can rent a stroller at the parks. You can even save a few dollars by renting them for the length of your stay, which means each day that you'll plan to rent a stroller while in the parks, you'll pay for upfront and you'll pick up a stroller at each park by showing your receipt. My concern with this, is it leaves you with little options other than carrying your 3 year old (or making him walk) once you are out of the parks (airport, resort, etc). You can find more information about renting strollers in the parks
here.
Another option is to buy another stroller. You might find the cost is justified depending on your circumstances and how likely your child will make good use of a new stroller. I have noticed that the deluxe Disney resorts usually carry a stroller in their markets for purchase. (I would need to do some more digging to remember the brand and cost, if this is an option you'd be interested in. Just let me know.)
One more option to consider, is renting from a third party service such as Orlando Stroller Rentals. I have seen many guests in the parks with these strollers, which can even be delivered to your resort. You'll have access to it in and out of the parks and these may be a more comfortable option for your son.
Hope this helps!