Hi.
I use Purina Flock Raiser for all my birds from day 1 through life.
You actually CAN feed a rooster layer... it just isn't ideal. And *could* (does NOT mean will) cause kidney issues in the long term. But it is most likely in birds that are predisposed to it genetically and also fed a large amount of treats. There are a few fear mongers on here that make it seem like your birds will die if you feed layer. Plenty of people have rooster that live a very long time even though they eat layer. It isn't my personal choice... but I don't believe in fear mongering.
You can either use starter or flock raiser for you chicks, finish up your layer feed and start your switch with the next bag for your current layers, if that seems convenient for you. Unmedicated starter or flock raiser will be fine for them and might even see an increase in health and overall appearance from the increased protein since feathers are made from 90% protein and it's amino acids. In addition... higher protein has been shown to give better hatch results, which to me says more nutrition in the eggs my family is eating, verses those who feed 16% protein layer.
Yes the terms become confusing... all flock, flock raiser, starter, grower, layer... I like chicks to have at least 20% protein. I know some all flock has only 18%. And anything beyond 22% is not really needed. If organic is important to you that will impact the choices you have.
So to wrap it up since I seem all over the place...

I would go with a Flock Raiser for the chicks and work on the switch for the end of your bag of layer feed for your current hens. And use flock raiser for everyone with the OS you said you already have available for the layers. Your OS usage will increase once they are switched over completely, but that's OK. Nutritionally and financially it all evens out. Even though the higher protein feeds often cost a bit more, the OS is usually much cheaper than feed. Your little's might sample the OS but they won't over indulge. My flock is usually integrated by 6 weeks old.
Hope your new arrivals are loud and hungry!
Shipping is hard, I would be sure to have some poultry nutri drench in the water and I add homemade electrolytes. I prefer that over the save-a-chick packets. And I also crush the feed crumbles a little extra the first week. But ANY supplement should be helpful.