To be fair, this did come across as talking down to someone (and like you've ignored the point completely).
I can totally understand the frustrations as I have issues of massively fluctuating blood glucose which no amount of basal testing and recording of what feels like bloody everything seems to be able to solve (food, carbs, insulin, activity, amount of sleep etc etc). Unfortunately though the chances are the pump is the best way to control the problem because there's very little injections can do about dawn phenomenon (unless you inject as and when it happens, but even that would be easier with pump).
It sounds like the best thing you could do is see if you can get more CGM monitoring, maybe for a month, make sure the food (incl fat and protein details) and activity is logged perfectly for that time period, and if there are still no patterns I'd agree with Spiker and say make a case for keeping the CGM- that way you can adjust your basal rates for the majority of the time and the CGM will protect you from the odd nights where you don't have DP.
Type One Diabetes since 2001, Coeliac Disease since 2003 ish, IBS, and on and off depression and diabulimia.
Current HbA1c 11.7%