I agree I don't think it's a liver dump, but one of the features of Dawn Phenomenon is a temporary increase in insulin resistance, so it could be that you need to raise your NovoRapid dose to compensate? My insulin/carb ratio is quite different in the morning compared to the evening, DP seems to have an effect until about midday with me.
Caffeine is reported to increase bg levels (by stimulating adrenalin), but I think you need to drink more than one coffee to get that sort of increase, unless you're an espresso addict of course.
Are you really taking 5 units of NovoRapid for 5g of carbs? That dosage would be enough to slaughter over 60g for me!
Someone else can pitch in about Levemir, I've been on a pump so long I can't even remember what my long-acting was called way back then...