First, the two definitions:
The trolley problem
The trolley problem is a thought experiment in ethics about a scenario in which an onlooker has a chance to save five people being hit by a runaway trolley by diverting it to another track, hitting one person.
The fallacy of dilemma
It is an informal fallacy in which the proponent restricts the options to choose into a few, say, two. It is a fallacy because the framing of the premise is erroneous.
Back to the trolley
In my view, the trolley problem is a false dichotomy (two options) problem that does two things. It forces you to believe that there are only two options – kill five or kill one. It then helps you to justify killing the one as one generous act to save five. And this has been consistently practised by political leaders, especially of the oppressor types, to push their malicious agenda whilst satisfying the collective imagination of the majority.
Dealing with the trolley
The best way to deal is to resist the premise. Why are there diversions or two tracks? Why are there only two tracks? Why is the onlooker not closer to the five so she can save them (by pushing or something)? Why does only the diversion switch work and not the stopping switch?
Reading
False dilemma: Wiki
False Dilemma Fallacy Examples: Your Dictionary
Trolley problem: Wiki
Trolley Problem: Merriam-Webster
Should We Trust Nature More than Ourselves?: Slavoj Žižek & Yuval Noah Harari