If you don't have plastic boxes inside the shipping box, or at least racks, I highly recommend double bagging in plastic bags inside the double box. If the strikes move around even a little, you will have plastic cases that have been crushed.

I had a guy tape all the strikes together in nice stacks inside the box and padded with bubble. There was just enough play in the stack, all the cases were in pieces with the strikes still nicely taped together.

I had a single bagged group burst through the side of the box, not the corner break.

I recommend splitting the shipment between 2 Medium Priority boxes or better yet, use a Large Priority Box for 350 strikes.

As Gene said, it will be very heavy. For the sake of the mail carrier, you should label the box as heavy, so they don't get a hernia trying to lift it.
Just my opinion.
Happy Strikin'
Peggy