Does the Moon phase still affect the visibility of the Milky Way even if it is not above the horizon?
No. The main culprit is the Sun. Lunar twilight is extremely weak. It essentially ends when the Moon goes below the horizon. See this answer at the physics stackexchange.
One problem is that this is the wrong time of the year to be looking at the Milky Way. The brightest part of the bands we call the Milky Way is the core, and that's best seen from May to October. Right now the core of the Milky Way is visible from the Northern Hemisphere for only the last few hours before sunrise, and it's not very high on the horizon.
There is another problem, however ...
The Moon doesn't even rise until 9:12am.
That's backwards. On March 8-10 2015, the Moon rises between 10:00 PM to midnight and sets between 9:00 AM to 10:30 AM. The Moon will be big and fat in the wee hours of the morning sky, which is when the core of the Milky Way would otherwise be visible.