I have quite a few Art Deco era furniture pieces. One thing I find with them, the wood veneer has a very distinct color not seen in many furniture pieces from today. Most times the hardware on them is that brass in a very bronzy color, often with bakelite or celluloid accents. On my pieces (which aren't a matching set), the bakelite is an amber color. I always pay close attention to those colors when choosing a wall color. The colors that you mention (firebrick, rust, chestnut) are likely to make your furniture blend in rather than stand out. Same for the Spanish poster (which I LOVE, by the way).
Here's some info about using a color wheel to relate wall color to furniture.
Using the Color Wheel for Wood Tones
In my home, I've had the most success pairing my vintage furnishings with green or lavender tones, both of which need a gray-ish undertone to look right. My favorite color was a Benjamin Moore color that's not made anymore but it was in the lavender family. It didn't look girly at all because of the gray undertones and the orangish furniture that drew attention away from the wall color. I looked on the B-M website and here's a color that I think would work, although on your monitor it may look different.
dusk to dawn 1446 Paint - Benjamin Moore dusk to dawn Paint Color Details
My second favorite in my long time of having this furniture was Restoration Hardware's famous Silver Sage. It's the color the walls of their stores are painted and comes in several depth tones. But I don't think those colors would complement your World's Fair poster which looks very blue on my monitor.
Silver Sage Paint Collection | Paint | Restoration Hardware
Also, in the green family olive tones would look good, too. It's a color that was popular in the era these furniture pieces were made. If you look at a real olive, you can see that the color has a gray undertone. And the pimentos they stuff them with is an orange-red. And olive trees are very Spanish.
olive tree 392 Paint - Benjamin Moore olive tree Paint Color Details
or
spanish olive 1509 Paint - Benjamin Moore spanish olive Paint Color Details if you want to go lighter and grayer.
I would think your metal accessories would look good sprayed in a metallic dark brass.
If you can remove a drawer of your desk, you could take it to a paint store and ask for suggestions. I've had good luck doing that in Sherwin Williams stores. Where I live the clerks there are always eager to make suggestions. My Benjamin Moore dealer is also very knowledgeable about color. They will be able to take into consideration the light you have in your room,too, which is important. BTW, B-M has great little paint tester jars that enable you to put a bit of color on your wall before you make a commitment.
Best of luck with your project.