It's worth noting right from the start that 3/16" plywood is very thin for the bottom of a 36"x39" drawer. You're correct in thinking that you will need to do something to prevent the bottom from sagging too much. But on the bright side, your drawer bottom doesn't need to support 50 lbs--or, more correctly, it cannot support a full 50 lbs without exceeding the capacity of the rails you've chosen.
The easiest way to strengthen your drawer bottom is to make it thicker. However, the obvious tradeoff is that, with a limit of 50 lbs on your rails, the heavier you make your drawer, the less capacity you have leftover for the stuff you want to put in the drawer.
Cedar has a density of about 23 lbs/cu.ft., and your drawer frame (sides, back, and front) is about 303 cu.in., or 0.175 cu.ft.
23 lbs/cu.ft. * 0.175 cu.ft. * = 4 lbs
So assuming your design uses an integral, inset drawer face, you have 46 lbs. of capacity available for your drawer bottom and whatever you're going to put in the drawer. If you add a false front, you'll have to account for the weight of that, as well. Once you decide how thick you want your drawer bottom to be, you can find the weight of a full sheet of that type of plywood, divide by the number of square feet, and multiply by the size of the drawer bottom. Subtract that from 46 lbs., and you'll have the maximum remaining load capacity.
Moving onto some of the remedies you're considering...
Put the rails under the drawer with guides at the edges
If I'm understanding correctly what you mean, this is not a good solution, for multiple reasons:
- All you're doing is effectively extending the support of the left and right sides by an inch or two on each side. The middle, which needs the most support, is still unsupported.
- If the drawer bottom sags significantly, it will produce a radial force along the sliding axis, so the slides won't roll as smoothly and may wear out prematurely.
A center-mount drawer slide will do more to support the bottom, but the bottom of your drawer will need to be thicker than 3/16" anyway in order to screw the slide to the bottom of the drawer.
Install wooden cross bracing in the drawer
You can add support ribs either on the underside or inside the drawer. If you're going to add them on the inside, you might as well just make the bottom thicker instead.
Make the bottom thicker, like 1/2 inch
A 1/2" bottom is more than adequate. 13/32 is borderline for a 50 lbs uniformly-distributed load, and anything thicker than that should support a 50 lbs. load just fine.
Don't do anything special but do glue the bottom firmly to the box
This will help to the extent that loading the drawer will produce tension across the drawer bottom rather than pulling the edges of the bottom out of the grooves. Unless it's under significant tension even when it is empty, the bottom will still sag--but it may not sag quite as much.
use narrower drawers instead
This would certainly work but you mentioned you aren't interested in this option. Another variation on this idea is to put dividers in the drawer and attach the bottom to the dividers.