Does "X is immoral" arguments beg the question?

What I am saying arguments that of the form of "X is immoral" when used against moral systems such as Kantianism begs the question.
Here's a example below.
Person 1: so you are a utilitarian?
Person 2: yes
Person 1: Imagine you are on a bridge with someone else with a track under the bridge. On the track is five workers, also on the track is a runaway trolley that is heading straight for the workers.
Person 2: What's the point of your thought experiment?
Person 1: To refute your system of morality. anyway, the trolley will kill all five workers if reaches the workers and the only way to stop the trolley is to push the person next to you onto the tracks and use the corpse as a trolley stopper. What will you do?
Person 2: I push of course since I produce more happiness saving 5 people than saving 1 person.
Person 1: But killing people is always immoral and since utilitarianism requires you to push, it must be immoral.

The only clear fallacy in this scenario person 1 is begging the question by saying that killing people is always immoral. Even if person 2 agreed to the premise that killing was always immoral, person 2 could state that any system being used in the same moral dilemma would necessitate some immoral action—either the immoral killing of one person, or the more (arguably) immoral neglect of saving five lives. Based on ones values, one can certainly argue that doing nothing and allowing 5 people to die is a far more immoral.

This could be a self-defeating argument IF a) person 2 agreed to the premise that killing was always immoral, and b) agreed with the premise that any moral system that requires an immoral action must itself be immoral. I am not sure if anyone would agree with those premises, however.