What you describe is exactly how my gastroparesis was discovered. Lows after eating, highs hours later when insulin was about gone. I was diagnosed less than a year when gastroparesis reared its ugly head.
Gee, diabetic stomach damage isn't too descriptive.
I didn't bother having a stomach emptying study because morning when the test is done is the least problematic time.
Gastroparesis makes managing diabetes a whole new frustrating game. I typically bolus 1.5-2 hours after eating dinner. I'm careful not to eat too much fiber, raw vegetables (though I still eat salad) & red meat, that takes forever to digest.