Poking around for a bit I found that the left mouse button is the [5] key on the numpad, and the right mouse button is the context menu key (which you also find on the mute key using [FN]).
For those that have frequent problems with the built-in mouse buttons and find that the accessibility options used to map them are unreliable, here's a simple solution: https://github.com/AyrA/MouseRemap
This repository contains an autohotkey script to remap the fake buttons to real mouse buttons. A readme is provided with details and instructions, but it's fairly trivial to use.
Don't forget to disable the keyboard mouse in the accessibility settings if you want to use this.