Interesting idea, I do not know how the 'etching' into acrylic process works nor remeber seeing such a photo so I cannot comment on that.
But, with acrylic, if the *edges* of the acrylic sheet are highly polished, light that is directed into an edge will travel right to the other side and illuminate the sheet from within. You see that effect on the display panels of some older electronic equipment, especially one made before LCD displays took over.
You might have to use more than one LED or use one at each edge, experimenting will tell you. The picture is, of course, behind the acrylic but in contact with it.
'Back lighting' is a bit of a misnomer there since the light would not shine through the back. LED's, with their narrow light beams, are not very good at evenly illuminating something from the 'back' side.
You would have to make a hollow picture frame to house the batteries which power the LED's unless you devise a way to use the picture hanging wire to carry the electricity to the frame. In that case keep it all low voltage and don't forget that you need a dropping resistor for each LED to limit its current.
Klaus