I did some searching on the Internet, and it seems that the Catholic Maronites use mainly Western Middle Syriac/Western Middle Syriac Aramaic as their liturgical language. (Even though they also seem to use Arabic (look under heading '(5) The Liturgy') and in the Diaspora sometimes even the local language - e.g. English in the US.)
So even though you are not asking about Biblical Aramaic but rather about one of its successors, this is still a 'dead language'. Obviously these are usually not taught like living languages with modern methods, and so it seems very unlikely to find a CD/DVD with "basic dialogues used in everyday situations" for you to listen to (and as a matter of fact, I didn't come upon something like that in my searches).
If my assumption is right that the Aramaic used by the Catholic Maronites is the same as/a variant of Classical Syriac (if someone knows for sure, please leave a comment!) then there are of course books teaching the latter. Some of these even come with CDs, but they seem to use mainly examples taken from the Peshitta (the standard version of the Bible for churches in the Syriac tradition) or from some other classical Syriac text. You also have to realize that all these books teaching Classical Syriac are in the traditional style with a strong emphasis on grammar etc. All this is obviously very far from what you have been looking for, but if your desire to learn this language is greater than your aversion to these learning methods then here are just a few links (mainly to Amazon):
1) Leshono Suryoyo: First Studies in Syriac I mentioned this book first, since it seems to come with a CD (but some of the reviewers complained of not receiving one, others mentioned it comes only with the Hardcover edition). Both the text and the audio can also be found on archive.org.
2) Introduction To Syriac This book comes without a CD, but I included it since it judging from the reviews it seems to be a much easier book to use for studying without a teacher than the first one. There is also a Key to Exercises but with very bad reviews.
3) The New Syriac Primer: An Introduction to The Syriac Language This book does have two 5 Star reviews on Goodreads, but I put it last since the publishing house's link for downloading the accompanying CD material seems to be dead.
The question remains of course if the language, and especially the pronunciation taught in these scholarly books is still the same as the liturgical Aramaic used today in Maronite Churches. Quite likely there will be some differences, similar to the differences between Classical Latin and the Ecclesiastical Latin of the Catholic Church.
To sum it up: No, there seems to be no simple way to learn Aramaic as used liturgically by the Catholic Maronites.
So finally, a last thing for you to consider: there is an online course in Surayt (also known as Turoyo), "a Neo-Aramaic language, traditionally spoken by the Syriac Christians (also known as Arameans and Assyrians) in different countries in the Middle East, and today by approximately 250,000 people in Europe. Surayt is a continuation of the ancient Aramaic language..." For the time being this course exists only in German, but an English version is supposed to go online soon.
