I also tested in a more stripped down desktop environment (openbox) and that didn't change anything. I went all the way back to 3.16-9 as well, but SSE wouldn't even load. I've tested using the following Proton versions with SSE on clean prefixes with no mods/SKSE installed.
You should see the banner along the top indicating ReShade loaded, but pressing the Home key will not bring up its UI after the main menu appears. All you have to do to reproduce is install SkyrimSE, install Reshade, and start the game. All three injectors exhibit the exact same behavior. As far as I can tell, the actual injection of effects is working- it's just the settings interface that won't come up. However the hotkeys DO work during the few seconds the Bethesda logo is on the screen. Basically, none of them respond to their hotkeys to bring up their respective GUIs after the main Skyrim menu appears. I've been trying to get post-processing injectors working in SkyrimSE and I've found an issue that affects every one I've tried, including reshade, gshade, and enbseries. I know this is really niche, but I wanted to at least document it somewhere other than random Reddit posts.