Completely theoretical idea:

1) Make a stencil out of metal. You can make it yourself by acid etching a thin piece of tin plate.
2) Insert the stencil into the lens of a flashlight so as to make a kind of gobo.
3) Press the flashlight against the paper and turn it on for just a second.
(Turn the light off before lifting from the paper!)
4) Develop. You will have your logo burned in black letters onto your picture.

This will take some experimentation to get it right but I think it could work if somebody was willing to tinker.

Two potential problems I can think of would be:

1) The stencil has to be pressed tight against the paper to get a clean edge on the logo.
2) There can be no light leaks in the flashlight or around the edge of the bezel that touches the paper.

However, if the kinks can be worked out, this would be the photographic equivalent to rubber stamping your name and logo onto a picture.