I say if renting a car is no problem for you then I would go with that option. We have always taken our stroller on the bus, and it is a pain. We take our own stroller because DD will take a nap while in the park. We don't rent one because we use it so many other places than just in the park. Like: while she's asleep in line while waiting for the bus, at DTD, around the resorts.... I personally couldn't get by without it, I especially couldn't carry her around all those places.
DH is in charge of the stroller on the bus while I hold DD. He stresses about accidentally hitting people with it while we get on. Once we are on its no big deal to have with us its just the getting it on part, its almost impossible not to accidentally hit someone. We also always wait to be the last people off the bus so that we don't also hit people as we get off. Once we tried to go through the back double door after people had exited and the driver actually yelled at us. Even when he was already driving away he got on his speaker and continued to lecture us. I about died of embarrassment. We weren't trying to sneak on, there couldn't have been even 10 people on the bus, we were just trying to make it easier on everyone. Oh well..... a nice "Please don't use the back doors" would have been nicer and I wouldn't have gotten the point just as well. A few times I did ask the driver prior to boarding if we could use the back door, some said yes and some said no.
I am sure it would be much easier with an umbrella stroller but our stroller lays flat which is perfect for napping and I don't think DD would nap in an umbrella stroller.
Wow, I rambled a lot, sorry!