Steam Deck natively recognizes the PS3 controller. Simply plug it in via USB, wait for Steam OS to recognize it, then unplug USB cable and it will work over Bluetooth without needing
Amazon.com: DualShock 4 Wireless Controller for PlayStation 4 - Jet Black [Old Model] : Video Games
Connect PS3 controller (DualShock 3) to Playstation 4 wirelessly - YouTube
Do PS4 Controllers Work on PS3? How to Connect a PS4 Controller to PS3
Do PS4 Controllers Work on PS3? How to Connect a PS4 Controller to PS3
How to use a DualShock 4 wirelessly with a PS3 | Eurogamer.net
Which Controllers Work With Stadia, PS Now, xCloud, and Nvidia GeForce Now? | PCMag