There's a thing like DP, but not DP that seems fairly common - a kind of "getting up phenomenon" which I definitely get and seems to affect lots of others too.
My BG is ok when I wake, but as I step out of bed, it starts to rise and, like yours, will often continue to rise through the beginning of the morning.
I currently deal with this by bolussing pretty much as soon as I get up and bolussing two or three extra units (I've pretty much worked out the amount using trial and error) on top of what I'd need for my breakfast.
I then do all my usual morning stuff and eat 30-40 mins later.
That seems to pretty much arrest the spike and set me up for the day.