John,
the reason to consider reinforcement strips in selected locations on the inside is that the strength increases disproportionately. You find that principle applied everywhere, from door blades to I-beams.
That said, if 3/8" is sufficiently strong, that's probably not worth the additional effort, although I bet that a hatch from 1/4" panels with 1/4" reinforcement strips at the correct locations could be stronger (= stiffer) than a 3/8" panel while being substantially lighter; that's because the reinforced areas would have a combined thickness of 1/2", and stiffness increases faster than linear with depth. The same effect means that the reinforcement strips don't have to be very wide to be effective.
If you instead of 1/4" plywood used a few lengths of 1x1's as reinforcement, the effect of the additional depth would be even more pronounced. Done correctly, the panels would probably bend far less than 3/8" plywood.
The one measure in which a 1/4" panel would show limitations is on resistance to puncture. However, I don't see that as an issue for a vertically attached hatch, unless you fear you are going to put your elbow through one

.
Of course, if saving weight was the primary concern, I'd go with a panel laid up from carbon fiber.
As you suggest, using a simple 3/8" panel might just be good enough for the purpose.