So many different issues here. What's best for them long term versus what you can do in an emergency can be very different.
The calcium in Layer is at a dangerous level for growing chicks, but that is what they average eating over time. And it depends on how much other things they eat. Could those chicks survive on Layer for a week without damage to their liver or kidneys? Probably. It sounds like you might go longer than a week. The longer they go the more the risk.
Chick Starter is generally higher in protein content than Layer. Different brands have different protein content so I can't say for sure what you can get. The higher protein gets them off to a good start in feather development and just growing. That doesn't mean they will die if they get a lower protein feed, just that they don't get as good of a start. And it's not just protein. Chick Starter has other nutrients they need. They could survive for a week on corn meal but they will not get off to a decent start. It could affect how they grow later even on a better diet, but corn meal isn't that bad in an emergency. But it needs to be corn meal, not one of those cornbread mixes with salt and other things in it.
In your situation I'd buy a small bag of Chick Starter so you don't hinder their development. When they leave, add that Starter to your other chicken's feed. Mix in small amounts so you don't mess with the calcium content they need for the egg shells. Or if you offer oyster shell on the side just feed it to them. It doesn't need to be wasted.
Many of us never offer medicated feed or give them Corid. Corid only works against the bug that caused Coccidiosis. If they are not exposed to your other chickens or the ground your chickens walk and poop on they should not be exposed at all. If you keep the brooder dry Coccidiosis generally isn't a problem. If you only have them a week or two, that's not long enough for Coccidiosis to develop into a problem even if they are exposed. I would not bother with the Corid at all.