Good question. I can see how this monthly occurrence would cause a problem in Edward and Bella's relationship. However, this is not an issue that Stephanie Meyer addresses. As you know, the Cullen family practice a unique kind of vampirism whereas they only drink the blood of animals and not humans. The urge is still there , they just choose to ignore it. For the most part, most of the Cullens avoid situations where they will be exposed to the scent of human blood. For example, Edward skips science class on the day they test for their blood types. On Bella's birthday, she cuts her hand on a broken vase and the others must hold their noses and leave the house until it is cleaned. It is difficult for all of them to have Bella as a part or their lives when things like this can occur. The only member of the Cullens who seem impervious to the scent of human blood is Carlisle who is a doctor.
This is addressed in the movie Twilight, when Bella walks into the classroom with a fan blowing on her. Edward's reaction is a clear indication that Bella was having her period. Edward due to this didn't return to class for a week. Her scent due to this was so intensified that he reacted by holding his hand over his nose and running out of the class. Throughout the movie his reaction is never to this level. I feel it is a clear indication of Bella having her period at that time.
He might have gotten stronger to the attraction when she is having her period as she would have more blood but this is a different type of blood then what he drinks straight from her veins so that may be why it does not affect him as much. Or he has gotten better at handling the scent of Bella's blood and Bella.
Good question!. I am going to guess the Edward pawns Bella off on Alice while watching her secretly from afar. Or he probably notices it but ignores it since he sort of built a tolerance for Bella's blood that time she got bit by James and he had to suck out the poison (this happened in book 1).
Stephenie Meyer was once asked this not long after the books were released- i mean, in a relationship with a vampire it seems like a major thing. She said that basically the blood from bella was "dead blood", easy to ignore because mostly it is not blood anyways. Edward spends so long ignoring bellas blood so that he can be with her so really, dead blood which is not attractive to vampires (no pulse etc) , it is not a big deal for him. The two bloods are different basically- one he wants the other he most certainly does not!