OK so since I’ve been asked a couple of times…
The process is basically this:
Cut your tray sides, assemble them, essentially making a hollow frame. I glue mine up, so I use corner clamps to keep things square while everything sets, and weigh the corners down to keep the entire frame flat. The end result is a nice hollow tray with no back.
Now I cut two layers for my bases. They should be made to fit pretty exactly on the inside of your tray frame. The bases have screw holes in the corners for rubber feet on the bottoms (see simple tray in the OP for pictures), but for now they are just convenient registration (alignment) holes.
You have choices at this point. I choose the slower, more precise method, but will touch on the shortcut at the end…
Remove masking (if any) from the facing surfaces between your base layers. (I actually only mask one side of one of the base layers, which I then put on the very bottom of the tray. This makes it easier to clean up later, no worries about glue).
I apply glue between the base layers, then screw them together. The screws do a great job of keeping the two layers aligned. Clamp and/or weigh it down, you want this to dry flat and well-adhered.
Once that’s set, I can then insert it into my frame. I am dead-set on not getting glue inside my frame, so I take it slow. Be sure you’ve removed any glue that pressed out of the seams between the base layers (best to do this while it’s drying, easier than removing fully set glue), it will interfere with your fit otherwise. Remove the screws.
First, a final fit test: I set the base down on a work surface, then gently try to slip the frame over. I’m going for a tight glue fit, not a friction fit. If it is a little too snug, I will take the base and sand the edges down a tiny bit. Be patient, oversanding can’t be undone, so take it slowly.
Prepare your work surface. Place something that can get glue on it on your work surface. I use wax paper with a layer of paper towel on top for some cushion and “give”. You should have your work area prepared now, because you’re about to have to move fairly quickly.
Once I am satisfied with the fit (this one went together on the first shot), then I get a brush and apply glue in a thin even layer all around the edges of the base. I try not to get any glue on the surfaces of the base (though your bottom surface should still be masked if you’re doing this step by step with me). You’re going for as little glue as possible while still getting a nice even layer on here, more glue is more cleanup. As such, it’ll dry quickly, so you’ll have to be even quicker.
Place the base on the work surface, bottom-side down. (You did remove the screws, right?)
Carefully slide the tray frame down around the base. This should have the effect of pushing a small amount of excess glue down, and prevent any squeeze-out of glue up into your tray.
I like to take a second to carefully wipe off as much glue as I can from the bottom at this point, glue is much easier to handle wet than set.
Then I put it back on the dead-flat work surface, weigh it down. I put weights inside the tray on all the corners and in the middle, then weigh down the corners of the outer frame. The end result should be a totally flush, level fit for the base into the frame.
Let it set, and you’re all done except for the finishing.
There is a slight shortcut method: You can glue the bases together and into the frame in one shot, and let both the bases and the overall tray set at the same time. Personally I think a lot can go wrong when trying to hurry – and especially when working with glue – so I tend to do it the way I described.
I don’t have a huge list of glues I’ve tried, but titebond classic has worked very well here. I would stay away from glues that expand (gorilla glue, etc), too hard to control where it ends up.
So, there you have it. Too many words for what is really a pretty simple process. “Glue it in” would probably have sufficed. 
EDIT: I suppose an even better shortcut would be to cut the base out of 1/4 material, but I don’t have any. Maybe I should get some, it’ll make the basing process a lot simpler.