2 Ashmoros after midnight

The gemara in Brachos 3b quotes a pasuk from Tehillim that says "kidmu einai ashmorOS" to prove that Dovid Hemelech got up at midnight before two complete ashmoros so each mishmara must be only three hours and not like R' Nosson said that each one is four hours. The gemara answers that it's no proof because he means that he gets up two ashomros before other kings. R' Yehoshua says on 7b that other kings get up in the third hour so, Rashi explains, that you have 6 hours of the night left plus the two hours until R' Yehoshua says other kings get up for a total of 8 hours or 2 R' Nosson ashmoros.

The problem is that we know that we pasken like R' Yehoshua that you can say Shma until three hours of the day because that is when kings get up but it means the end of the third hour and not the beginning. So that means that you have 6 hours from the night plus three from the morning for a total of 9. This gemara, therefore seems to prove that Rashi is correct that R' Yehoshua doesn't mean until the end of the third hour but until the beginning of the third hour so you only have two hours each day before sof zman krias shma. Most of the other Rishonim though learn that R' Yeshoshua means until the end of the third hour so how do they understand this gemara?

I was thinking that you could answer the question because all we have to prove is that there are at least two ashmoros after Dovid Hamelech wakes up. The gemara assumes that ashmoros means at least two so you have six hours in the night plus two in the morning to get two complete ashmoros. There happens to be another hour in the morning also but that doesn't concern the gemara as long as we have at least eight.

The Pnei Yeshoshua suggests a good answer: R' Yehoshua holds that you can say Shma until the end of the third hour because kings wake up during the third hour. Some wake up at the beginning and some wake up at the end so zman kima is until the last kings wake up but Dovid Hamelech wanted to say how much better he was than even the best of the other kings. The earliest any kings wake up is the beginning of the third hour so Dovid Hamelech is saying that he wakes up two full ashmoros before any other king.