What are some strategies for handling the transition between different-aged beans in a coffee shop Espresso grinder? Suppose my grinder's hopper contains beans roasted 14 days ago, and the hopper is almost empty; suppose further that I have a full bag of 5-day-old beans, ready to go.

If the new beans are added to the hopper on top of the old beans, then there will be an abrupt change to the coffee's "dial in" once the transition between the old and new beans is reached. This is a drawback in the commercial setting; it would be hard to keep track of where the transition is, so some customers could receive improperly dialed espresso.

The dial setting will also be affected if I wait until the hopper is completely empty; the grinder will not grind consistently if the hopper is underfull.

The final option I can think of is throwing away the old coffee beans to make way for the new ones. This seems to be a waste of money. Any tips?
You will just have to find some form of compromise. I think the best way would be to empty the Hopper and then fill it with the new beans (also sometimes you might want to switch up the espresso?). You should figure out how your grinder behaves once it's getting more empty and with this knowledge you can quite easily adjust the extraction such that the quality will not decrease too much.

The most likely case is that the dose will decrease slightly. You could put a scale next to the hopper and if you feel like your espresso is running faster, check if the dose changed because the hopper is getting empty and adjust your grinder accordingly. I think checking the dose and grind occasionally throughout the day is a good way to up your espresso game in any case. It helps me learn a lot about the coffee and setup. There's tons of variables, like humidity and temperature that will change the behavior of the grinder.