Did you connect the USB connector of the CAC-1332? Maybe it’s not getting enough power. Can you connect the USB connector of the CAC-1332 to a dock or hub?
Are you using a USB-C cable with the CAC-1332? Maybe it doesn’t work with a Thunderbolt cable.
When the Mac mini detects a display, it will appear in the Displays system preferences panel. If it doesn’t appear there, then you cannot test different resolutions.
If the display can get detected, there probably will not be a 5K option, since the LG UltraFine 5K display does not have a single tile 5K option in its EDID and there’s no way on M1 Macs to create a custom timing like on Intel Macs.
For firmware update, I guess you need to ask Club-3D tech support if the adapter is supposed to be able to do width greater than 4K or not. If they say it should be able to do width 5K, and your adapter cannot (like mine), then maybe there’s a firmware update.
Club-3D doesn’t say what chip is used. SIIG and Delock have similar adapters that use the LT6711A. It has some USB-C specific features, so maybe that means the CAC-1331 uses a different chip? Anyway, the info for the LT6711A doesn’t say it supports 5K width.
The Lontium website at http://www.lontiumsemi.com/product/list_1849.html has a LT6711GX which can convert HDMI 2.1 to DisplayPort 1.4 with USB-C. It works with DVI1.0, HDMI 1.4 – 2.0b (18 Gbps) and HDMI 2.1 (32 Gbps). It supports resolutions up to 8K 30Hz (or 8K 60Hz with DSC). So I guess if the LT6711 can’t do 5K width, then maybe the LT6711GX can do it.