How can I subfeed a single pole switch from a 3-way switch to control a fluorescent light?
That's not so clever. The problem is if the washer light is ON, and the utility room light is OFF, then flipping the farther 3-way will turn the washer light OFF and the utility room light ON, necessitating a trip to the other 3-way to put it right.



You need a major reassignment of wire functions in the 3-way circuit. 


Ground is ground
White is neutral, always
Black is always-hot, always 
Red is a digital control line (or it can be a relay actuation signal line)


The 3-way box that feeds the lamp gets the "master" smart switch (or a 3-way and the relay).  The "remote" smart switch goes in the other 3-way box.  Some smart switches use a data-communication wire, that's what red is for. Others use radio or powerline-induction signaling, in that case red is unused.  

Now there's always-hot and neutral in every box regardless of the topology of the 3-way system.   Now you can add the second 1-way switch in the normal fashion for adding a 1-way.