When writing an empathic/monologue-like piece of writing, would it be acceptable to switch between first and second person?

The decision of whether to use first- or second-person depends on the type of writing.
A monologue is usually a dramatic piece of writing. From my experience, most monologues are always first-person. In a traditional monologue that an actor would use for an audition, the actor/character would be speaking from their own perspective and relating thoughts and emotions based on personal experience. To make the piece relatable and personal, the first-person "I" is recommended. The only reason I can think of for using the second-person "you" in this theatrical type of monologue is if the actor/character is talking to someone and attempting to explain events from the other person's point of view. For example,

Actor 1: "You would brush your hair, put on make-up, and act like nothing happened. You would leave the house for hours without a care about anyone else's feelings."

In this context, events are still being told from the actor's point of view and no one else's. Since the actor is the main character as opposed to the reader, this could still count as first person, but with the use of "you" it can also count as second person.
Getting outside of the theater, I believe it's perfectly fine to use both first and second person in a piece of writing that is similar to a personal essay or narrative. It depends on the tone you want your piece to have and whether the piece is informal or formal. Sometimes, if the piece is informal, you can get away with writing any way you want—as long as you have a distinct purpose for your writing and as long as that purpose isn't confusing to readers.

According to the Encyclopedia Britannica, monologues can be broken into three categories: soliloquy, interior monologue, and dramatic monologue. In all three categories, we can easily envision how the second person might allow for dramatic and tonal possibilities not otherwise available. As to an empathic piece of writing, the very basis of empathy, as cited in the scholarly article below, is contained in the pithy second-person statement "I feel your pain."
In formal writing, tradition typically forbids the use of the second person. But if your piece of writing is informal in nature, it should not be forbidden. Examples of informal writing might be anything fictitious or any type of non-fiction form where a lot of latitude in presentation is expected or tolerated—for example, memoirs or books that combine humorous anecdotes as illustrations. This is in contrast to academic articles, which require much more formal restraint. Monologues are typically far removed from the world of strict academic writing, so I feel quite safe in reassuring you that there is no specific reason to preclude the judicious application of the second person, unless your assignment specifically dictates otherwise.
http://www.dlls.univr.it/documenti/Seminario/documenti/documenti336895.pdf