I honestly think this style is do-able. I have experimented with this style myself before (though none of which I like enough to share).
Font, style, sentence structure, spaces, and the use of italics to emphasize certain moments become crucial. Simply because there are no use of adjective to state how a sentence is exchanged during a conversation, it makes the interpretation of the mood a true struggle. So it's quite an art to write in this format and I am loving that you are taking full advantage of that. Even the use of the messaging format to make it as similar to an actual messaging platform. It's great.
As far as the plot goes, I understand everything that's going on so far. It's not difficult to follow and I'm glad you stayed true to the experimental format.