Just glue on a few cross braces (ribs) of the same acrylic material a few inches wide on to either the top or the bottom side of the lid. I have 3 foot by 3 foot chambers with full lids that have two 2 inch cross braces for a SOG that support the weight of the mature plants just fine. You could use thinner sheet for the lid by using the ribs but that would mean gluing up ribs or buying two thicknesses of acrylic. The thinner acrylic used as ribs would twist (buckle) to easily. Just consider it like building floor joists or beams with thin flooring glued to them. The sheet would only need to be strong enough to support between the ribs. If your really worried or plan on growing say 4 foot tall plants then place the ribs every 6 inches.
From a national manafacturer of acrylic plastic:
A practical rule to follow in outdoor installations is that the frame depth should be 1/4" plus 1/8" per running foot of acrylic sheet in length and width to provide for contraction.
(deflection induced shortening)
As a general rule, any horizontal outdoor glazing for skylights, etc., should be 1/4" thickness for minimum deflection with maximum unsupported span of 36".
http://www.rplastics.com/plexdesign.html
This means as use for a horizontal skylight (basically what you are building) that is 36" a square (maximum recommended without some form of support) the the plastic will deflect enough in the center so as to require the edge frame to be at least [1/4 + 4 x (1/8" )] = 3/4" . This means a 48" it will bending enough so as to draw down almost to 47-1/8". A few cross braces will allow for 1/4" to span 48"
and minimize the deflection. I would just make the lid 48" square and the size of the outter edges of the box say 47" square and glue on an outter rim of 1/4" acrylic and three cross braces on a 48" square lid. I made my nominal 36" boxes 35-1/2 inches wide on the outside dimensions with 1/4" thick 2" deep lips and two 2" deep cross braces. With mature plants the lid is tight enough due to deflection that when done again I will make the boxes 1/4" smaller (35-1/4 inches), so I think 47" would work fine for a 48" lid.
If black acrylic is not readily available just use clera. Paint it first with black and the white over the black.