It’s totally OK to work out while you’re fasting. Many people do it and even prefer it. I would try it and see how you feel — it might work great for you.
I wonder if something like 12 PM-8 PM eating window might work for you. That way you can have dinner after your early shifts, and lunch before your late shifts, plus (I assume) take a meal break in the middle of your shift.
Keep a journal of all calories you ingest. Then wait a few days, and look it over with a clear head. Mark the stuff you should skip, try again.
Have a nutritionist look a the journal if you can, too.
I do 2-10 PM eating window as well and had no issues working out in my fasting window while on vacation. In fact, I was doing 2 a day workouts with weight training in the morning and running in the evening.