Would it be beneficial to take 1.5 - 3 mg of melatonin in the middle of the night if I wake up and have trouble falling back to sleep?

NO. But you may try a higher dose. It is important that you take it 1-2 hrs before sleep, not at bedtime, for it to work the best. Please give it 3-7 days before you call it quits and try something else.
Sleep. Melatonin works better prior to falling asleep. You can try that dosage for a week, if it doesnt work then you might have to increase it to 5-6 mg. If still not helping then speak with your physician.