the poor mans solution is to use your standard rip or combination blade. Set your fence for the bottom of the dado to the bottom of the draw width. Make your cut on all sides. Reset your saw fence for the top of the dado to the bottom of the draw and run all sides again. It a two pass process for 1/4 ply but it is faster than installing a Dado blade or set in the first place and a lot faster than waiting for a delivery of a new router bit.
I don't have a router table and so I don't think router first.
I don't know for sure but try Amana router bit CO. They may have a bit that will do just what you want in one pass. If you are not doing alot of them (or will never do it agian) make two passes with a router bit smaller than the thickness of your materiel. Check Finewoodworking.com for info on how to do this. If you dont have a router make two passes with a thin kerf table saw blade. That will probablly be the easiest. Or use a different materiel for the drawer bottom. Be Safe!
You should have a thin, tapered cutoff floating around the shop. Make a tiny wedge and us it to fill in the difference. The bottom will tighten up and you'll never see it. Use razor knife to cut off excess and small drop of glue so it'll never come out.
Simple is good. Simple is fast. I've seen that technique used on older furniture to lock the bottom in so you don't have to put a nail into the bottom to lock it to the back. Since I like to avoid metal in all my projects, I notice things like that.
K.I.S.S. (keep it simple stupid) is not a bad motto.
Simpler yet, and I do this often, Turn your drawers upside down, toss (or place-if your not a good shot)a hammer or something weighty enough to keep the drawer bottom weighted down. Now run
a bead of clear silicone in the gap, let set and done. It allows for expansion/contraction, is easy to remove if needed. No chaseing bits of scrap or tedious alignments on the T.S. Only drawback is you need to give it ample time to set up.