The choice really depends on the specific circumstances of the story and what exactly it is that you are trying to argue "isn't very realistic." Problem is a much more general word for any undesirable issue or situation that is in need of resolution, whereas dilemma implies that the resolution of the issue/situation involves the specific choice between at least two alternative solutions, the selection of either of which is for whatever reason not optimal and therefore makes the choice itself difficult. Thus, another difference between problem and dilemma is that the latter implies that at least two solutions to a problem have already been envisioned, an inference that is not necessarily--and is usually not--true when one speaks only of a "problem."