Former Optician here -
The thickness of your glasses is determined by the curve of the lens needed, not whether it is a bifocal or not. Bifocal lenses and single-vision lenses should be the same thickness so long as they are exactly the same Rx.
"Far-sighted" lenses/Rx's start with a + (ie +.50, or +3.75) and will be thicker in the middle, making the eye appear larger. "Near-sighted" will start with a -, like -1.25 or -4.50, and the edges of the lens will be thicker than the middle. OP didn't specify which Rx her son has, so we as posters have no idea which lens her son will have unless she can clarify. Bifocal lenses can be made from both + and - Rx's.
I think most, if not all, optical manufacturers require children to wear polycarbonate lenses for safety reasons, because they are stronger and more shatter-resistant. These lenses are inherently thinner and lighter than plastic, or the glass lenses some people may remember from their youth lol
Progressives can be make from any lens material, and are actually a trifocal-ish lens since there is an intermediate distance. These lenses "progressively" (gradually) switch from distance to reading, eliminating the need for a line. This would not work for strabismus because the lens' (specifically the bifocal part) job is to draw the eye toward the optical center of the two focal points of the lens - this is part of the vision therapy that the bifocals are needed for in the first place. Kids don't really "outgrow" this condition, rather, the bifocals retrain the eye to focus where it needs to focus by strengthening the eye muscles. The PP was also correct - there needs to be a certain # of millimeters from the optical center of the lens to the base of the frame so that the area of progression isn't cut off, making it very difficult to fit a child-size frame. It *can* be done, the frame just has to be long enough to fit the progression.
Transition lenses don't get 100% back to clear when indoors, so keep that in mind - you will looking at your sons eyes through tannish-colored lenses. I'm personally not a huge fan of Transitions because I don't like the look of the constantly-tanned tint, but some people seem to like them. It's personal preference. I don't think that transitions will "hide" the bifocal any better than clear lenses will.
Hope this helps....if anyone sees anything incorrect above, feel free to chime it...it's been about 13 years since I've last worked in optical, so I'm not sure I'm completely in the loop on the newest recommendations and innovations!