The advantage of 16 bit is that it divides the levels between full black and full white into many more intervals. This keeps gradients smooth and avoids problems where multiple adjustments can lead to individual steps becoming visible as banding in the image.
Many print drivers work in 8 bit, but that is not really an issue as it is right at the end of the chain so no further 'steps will be introduced.
So my advice would be work in 16 bits and save the master in a format that uses 16 bit and keeps layers - so psd, psb or tiff. Print using whatever driver is available, if you printer and driver has a 16 bit option check it but you may or may not see any difference in the print. Similarly if you need to make a copy to send onwards or for web use, use Export to make a jpeg for that specific purpose (which will be 8 bit).