After further testing we found out that the present dropping phase is determined the seconds before the song starts. The game will autosave at the point it's generated, so...
When I did the initial tests, I rolled back to the beginning of the week, not the beginning of the song - It was a mistake in the testing I was using, although if you do the exact same actions, you WILL get the exact same result, which left me to draw that conclusion.
Once you lock into the song (Namely the moment it autosaves prior to the song load) the results of the drops are locked. Incidentally, this prevents 'p-drops' from being wasted if you lose power etc - the drops are already calculated in at that point, if you resume after quitting.
(So basically you could do some control and time dance to alter a 'bad result' but in all honesty, there's no way for you to know what magic combination is required to force a gold present.)