Sorry, With the high being 2-3 hours after a meal that would indicate a bolus needing to be corrected.
As I understand it you are-
Bolusing for meal around 8pm
High at 11pm
Hypo at 3am
High in morning..
This to me indicates:
Evening meal is either being:
1) too little insulin... (We do not know your levels before the meal, so this information would have been useful, but I have assumed that your level was less than 8.0)
2) your bolus injection is not being given at quite the right time for your meal that you are eating;
3) that your meal bolus may not be calculating to take in to account a need for fat and or protein consideration
Then your hypo's at 3am indicates that:
1) basal timing is incorrect (we do not know what time you give this basal injection)
2) the basal dosage is too high
Your morning high level indicates:
1) dawn phenomenon (do you ever test at this time) if you have a lie in? ) (does this high occur at this time if you do lie in)
2) over treatment of the hypo at 3am
3) have you always had higher levels in the morning?
4) depending upon the basal you are using, it may not be lasting 24 hours, but this would also depend upon what time you are taking the basal or if you are split dosing it.
And finally: it does sound as if you need a pup where you can fine tune your nightly basal rate insulin. However, if you could answer any of the questions above it may help to eradicate some of the swings you are having...