Hello, anon! :)
Oh, no. In fact I really like K Project related questions! So, you don’t really bother me at all. :)
It is true. He was shot at 23:45 on 7th December. GoRA didn’t state his death time, just the time he was shot.
Although, It is very possible that he died at midnight or around that time on 8th December, since after he was shot he called Kusanagi and waited for him and Yata to arrive where Totsuka was laying on the ground, covered with blood. Then, the anime scene takes place.
Also, In K Memory of Red chapter 15, we see Kusanagi talking with Suoh Mikoto on the phone for him to not get there. At the moment of the call, according to the anime, Totsuka was already dead in Yata’s arms. Anna comes to the King’s room and asks what is wrong only to be told that Totsuka died.
After that, to his despair, Mikoto Suoh becomes covered in flames and Anna runs up to him and hugs him. On a bedside table, there was a clock with a red rose where it showed that it was 00:07 on 8th December.
So, Totsuka probably died around midnight, which makes Anna’s Birthday a sad day instead of happy one, marked not for her birthday anymore, but for Totsuka Tatara’s death. That’s maybe one of the reasons why HOMRA members can’t wish her a “Happy Birthday” anymore.
That was the day HOMRA, Anna’s family, started to break apart.
It is extremely sad to think that Anna lost her parents and received a new family only to in her 11th Birthday see her new family starting to crumble away.
I hope this meets your answers. :)