Jumping in a little late to the party here, as everyone (and your doc etc) has said, the nurse it plain wrong. Insulin very basically just unlocks the door in the cell membrane to allow the glucose to enter from the bloodstream to be burned as fuel by the mitochondria. It doesn’t need slices of bread to “work on”. All it does is hoover the glucose out of your blood and shove it into the cells.
DKA is caused by an unholy trinity of no insulin, high BG and dehydration - not insulin being injected as a correction dose with no food. The very worst would be a hypo if you got the dose wrong.
Insulin doesn’t act on fat and you can’t make glucose from fat - just carbs and protein.
I regularly inject without food - Small corrections of just a half or a whole unit to nudge my levels into line, and a bigger shot foe the dawn phenomenon. As long as you keep track of how much insulin you have sloshing around (I use the bolus calculator in the MySugr app), even stacking small doses shouldn’t be a problem, although it’s quite an advanced trick that not all health professionals like us to do.
Sounds like you’re doing all the right things