Depends on where you get the school bus from but some places dont maintain them as well as they could. Your also going to have to make a door ig enough to fit a bed into.
I assume a lift equipped van is a full size van, I don't think I've seen any of the smaller ones equipped with lifts so its most likely good to pull a trailer. Sounds like the hospital bed bound person is able to travel via wheelchair in the van so that helps, you can get a small trailer and roll the bed up the ramp into it and hitch behind the van and go. a small enclosed trailer where you could sit in one side or stand partly up could hold other supplies like you say in the back of the rental truck.
There are more issues with a rental truck, first is availability, if a situation is bed enough that you need to leave then chances are a bunch of other people are going to need to leave as well and there will be some of them wanting rental trucks as well. You have to think about there being 200 people wanting the 100 trucks in the area and thats even if the rental place staff hasn't aleady left. I think it best to not rely on being able to get a truck.