figure-eight orbit, making one sun dominant at a time to reduce the difficulties
Yes, being able to turn one sun "off" at a time vastly reduces your headaches, and your story has sufficient "magic" / "science magic" to support it.
You still have a minor issue that two suns will want to orbit themselves (If they were stationary with respect to each other, they'd join up, and who needs that headache?). As a result the planet would have to travel faster on some parts of its orbit -- to stay ahead of the creeping sun. In effect your planet's orbit itself would have to be orbitting to stay out of the path of the sun. Now, the planet moving its entire orbit is possible if you introduce mass that will upset the balance. But of course, the question becomes how is that mass moving around to move the orbit where it needs to be.
Fun stuff!

