Can you extend shelf life by reheating?
What you're concerned with when it come to food spoilage is the toxins bacteria produce, not the bacteria. While heating it will kill the bacteria, the toxins are typically proteins which will not be destroyed until your food has been turned to charcoal. Heating it will thus not increase the shelf life. On the contrary: heating and the cooling means your food has been in the 'danger zone' for longer than if it was refrigerated the whole time.

If you want to increase the shelf life of a dish like you described, freeze it. If frozen as soon as it comes out of the oven, it will last for several months in a normal freezer. The sooner you get it frozen the better for extending shelf life.