Let me clarify this question as much as possible because this will be somewhat of a 3-parter question.
The Bladesinger's Arcane Strike level 3 class feature says:
When you use a bladesinger daily attack power on your turn, you may make a melee basic attack as a minor action.
So how I understand it is "You can make 1 melee basic attack as a minor action on your turn when you cast 1 daily attack power."
This would be a correct assumption, right?
So the other question becomes:
If I fire a bladesinger daily power, then make a melee basic attack as a minor action and haven't used my move action, then if I action point and fire a second daily attack power, would I be able to turn my move to a minor action and use it to do a melee basic attack a second time?
Finally, taking into the account above of whatever answer, what if I fire a bladesinger daily power and then action point and fire another bladesinger daily power? Would I get only 1 melee basic attack as a minor action due to never using my melee basic between the daily powers or could I do 2 melee basic attacks as a minor action each (assuming both was after the 2 daily attacks)?