Reddit mentions of 3.5mm to RCA Camcorder AV Audio Video Cable, 3.5mm Mini 1/8" TRS Male to 3 RCA Male Adapter Cord 4FT

Huh. That's a cute little mixer. It might actually work! I didn't realize that the connector was on your controller. It sounds like it's a TRRS connector. Do you use the mic? If not, your solution should work. If you do though, you'll need to break the TRRS out into individual mic, R, and L channels before the mixer, then combine them back together again into another TRRS jack. One way to do that is to use TRRS to RCA adapters like this one: https://www.amazon.com/Wpeng-Adapter-degree-Right-angled-Stereo/dp/B01LAJTTBU

and this one: https://www.amazon.com/Camcorder-Audio-Video-Cable-Stereo/dp/B07V2VZ7KP

You would connect the mic straight through, then run the left and right channels through the mixer. To do that, you'd use TRS (not TRRS) cables like this: https://www.amazon.com/Hosa-TRS-202-Insert-Cable-Meters/dp/B000068O1J Make sure to note the difference in sizes between the 1/4 inch and 1/8th inch TRS and TRRS jacks.

Speaking of, FYI, if you go with the mixer you linked and don't use a mic, you'll need two 1/8th inch to 1/4 inch TRS adapters like these (this is a five-pack): https://www.amazon.com/Cable-Matters-5-Pack-Plated-Adapter/dp/B00PIWB2SO/