No, as ATmegas don't have a barrel shifter. Also that 4 bit chunk in the middle doesn't do you any favors if you want a neat solution like shift and mask in a loop.
If the hardware design isn't set in stone, your best option would be to rewire so each 6 bit chunk uses the same bits of each '595, preferably LSB aligned, then you only need to split and mask in the two halves of the 4 bit chunk into the high bits of two of the bytes and mask in the five individual bits.