Baking – Question about chiffon cake

Well, the obvious answer is to buy a metal cake pan. Otherwise:

1. You could try putting a jellyroll or cookie sheet under it. Then grab the cookie sheet to rotate. Beware that depending on what the pan is made of, it may affect browning especially on the bottom of the cake. Adding some insulation (e.g., parchment) may help.

2. Plain aluminum foil may work too. That'll slide fairly easily on the rack, and again you can grab it easily. Especially if you pull the pan out of the oven, aluminum foil surrounded by air will cool quickly, and you'll be able to grab it without the inflexible mitts.

3. Similarly, a sheet of parchment paper under the pan. Same principle as the aluminum, except even at oven temperatures you can grab parchment briefly.

Chiffon cake, like it's cousin angel food cake, is mostly air. A big pile of protein bubbles stiffened with a little starch.

One very important step is not reflected in your recipe:

When the cake is completely baked the proteins have set and the starches have gelatinized but the starches are still very soft. The cake won't have its firm structure until the starches have been able to cool and set.

All recipes call for inverting the cake immediately when it comes out of the oven. The cake is allowed to cool, inverted and still in the pan, for a good hour to ensure the starches have set.

Special pans with legs or a long tube center are used for this:

Notice that in addition to the feet that it is a tube pan. This is important because the interior of these cakes is very insulated and won't be able to cook completely before the outside is overcooked.

When my angel food cake pans were packed I had good success using a pot with a mason jar in the middle. I was surprised it came out perfectly.

TL;DR -
Use a cake pan with a tube,
Invert the cake right when it comes out of the oven and let it cool completely.

*Note
Catija is correct that these amounts look way too small for a full cake. If you are trying to bake that in a full sized cake pan it could possibly be over rising and not have enough structure to support itself when it comes out.