I would assume that since you have legal permission to show the films you should be able to talk about the fact that you're showing the movies.
But to answer the bolded question:
[A]m I allowed to use the movie poster in my flyer?
I'm not entirely sure, you might know better than us in this case. While I would never encourage anyone to break the law and use images unlawfully, you are using this for a free, public, and most important legally-sanctioned event. You are not selling their posters, nor are you making money from showing the movies or posters. In fact, the posters themselves are promotional items for the movie itself. But, I'm not a lawyer. So don't take this to heart.
Strictly speaking however, it may still be illegal to reproduce the posters, even on your own fliers. (Though it could be possible that it's covered under Fair Use).
In determining whether the use made of a work in any particular case
is a fair use the factors to be considered shall include—
purpose and character of the use, including whether such use is of a
commercial nature or is for nonprofit educational purposes;
nature of the copyrighted work;
(3) the amount and substantiality of
the portion used in relation to the copyrighted work as a whole; and
(4) the effect of the use upon the potential market for or value of
the copyrighted work.
Source: copyright.gov : Section 107
Honestly, you should consult with your client, as they obtained permission in the first place and may have the answer or have the means to get an answer.
However, you are allowed to use the title of the movie:
Copyright does not protect names, titles, slogans, or short phrases.
In some cases, these things may be protected as trademarks. Contact
the U.S. Patent & Trademark Office, 800-786-9199, for further
information. However, copyright protection may be available for logo
artwork that contains sufficient authorship. In some circumstances, an
artistic logo may also be protected as a trademark.