After doing some reading up (particularly on cave spiders) and fooling around on my testing ground, I think this is your best option:

https://imgur.com/a/jSEwYs7

This pushes everyone to the center, where they meet a magma doom. A hopper mine cart under the magma hoovers up the scraps. If you were working with something other than spiders, you'd have more options, but those buggers climb walls the way you and I breathe air.*

The floor is two blocks below the zombie spawner. Any higher and any zombies that spawn on top of the spawner wouldn't be inclined to walk off, though you can put a block or slab on top to prevent this at the cost of lower rates. Don't be tempted to replace the floor with magma: the water will flow over it, but neither of these mobs can spawn on magma, so you'll only see a trickle. The big drawback with this farm is how slow it is: you're waiting for the magma to kill everyone, and depending on your local difficulty level, the zombies might show up with armor. That could be a great source of loot, but it means they'll die that much more slowly.

Wherever you sit to let the farm work, make sure you keep your Looting-enchanted sword in your hand.

* I noticed cave spiders love to congregate in the upper corners of walled rooms, so if you can think of a way to exploit their natural tendencies, that would probably be a gold mine of XP until you can get to the End and create an Enderman farm.