There's a flaw in your friend's suggestion: Leaving Doctor Strange alive couldn't mean he would have to die a natural death. He could reverse his ageing processes to become young again and again (while screwing Dormammu). And, he had no reason to hesitate because it wasn't his dimension (no breaking of natural law) when he had already used it in his dimension.
In case of his natural death, he would have returned again because time loop was supposed to restart upon his death (which should be any death).
However, I still believe that there was a way out: All Dormammu needed was to knock Doctor Strange unconscious. Just before the time loop scene, when Doctor Strange knocked down to ground, the time reversal stopped. It means that Doctor Strange was consciously controlling the Eye of Agamotto. Knocking him unconscious should have worked. In fact, in the last loop, when Doctor Strange was just knocked, he did saw the green holographic rings over his arm. I believe, he saw it to make sure that the time loop was still ON.
Conclusion: Doctor Strange just fooled Dormammu by making him believe that there wasn't a way out when in fact there was an easy way out. But, Dormammu type powerful creature might not understand it which seemed to operate as binary switch: Either kill or don't kill.