Given the nature of the content, it needs to be serious. On the other hand, the comedy therein provides necessary relief from intense suspense.
Enotes provides a very good commentary about what the genre might be. Even as you are struggling pinpointing the genre, that has been the case for literary experts too.
In fact it has been referred to as a "kitchen-sink" drama, meaning it has a little bit of everything!
Likewise, it was compared to Shakespeare's ability to create the comedic tragedy. Read the article, there are 3 really good paragraphs to give you ideas.