I had the same issue on Nobara KDE which is the same as Fedora KDE. The issue was with bluez the bluetooth api, version 5.69 is broken. Either you can downgrade it or wait for an update. To downgrade, use the command line "sudo dnf downgrade bluez" to downgrade bluez to 5.68 which is working for me.

I'm assuming you've already verified that other bluetooth devices work as expected.

A different distro (e.g. Debian Stable) might help narrow down the problem, as their kernels tend to be built with different options and their packages sometimes have different default configurations. Be sure to look for helpful messages in dmesg and syslog/journalctl.

After that, the next thing I would try is a USB bluetooth dongle. Some bluetooth modules don't work well with Sony's controllers, and it's possible that the one on your motherboard is among them.

For what it's worth, same issue happens to me on Kinoite as well. On an AMD board. With AMD 6E/Bluetooth.

My wi-fi works fine and my Bluetooth works fine for audio/headphones so I hadn't realized it was an issue until I got a new dualsense on sale a few weeks back and have been playing corded this entire time 😅

Are you trying to pair the controller while you have a pair of Bluetooth headphones already actively connected? The reason why I ask is because I've ran into issues with the AX200 when attempting to use both at the same time. Either the Dualsense would stop working or it would kick my headphones into the lower bandwidth HSP mode. I ended up having to use a separate USB bluetooth dongle just to get the two to play together nicely.