I see, on this forum many have had turn signal damage, and suggestions of what to do have been offered. Mine's a little different, because of the evolution of the 310.
I dropped my 2022 (I don' wanna talk about it - lol) while doing very low speed tight turns. That damaged the bar weight, and snapped off the clutch lever at the molded-in break point, as engineered. I took a nice, photogenic flight and hit the ground rolling along and snapping up onto my feet, as if I'd been a trained gymnast. Pretty good for a 76 years old porker (applause here, please). The helmet didn't touch the ground either. No tanky touchy, either. I thought the lever and bar end were the sum of my troubles.
At home, grinding, then sanding the lever end smooth, I somehow noted the rear right turn light dangling. A closer review showed that the wires had been broken, too. "No problem", I thought, "I can splice 'm". Then I saw that the end of the stalk is solid. It's not a hollow, pipe-like thing. The wires are very thin and one is flush with the end of the stalk. That leaves no room to solder or splice. May I can just pull it out or open the light or stalk. Nope - Seems to be one solid unit with the clear lense fused to the black surrond, with no cracks / opening lines visible.
The 'nub' or tapered part that fits into the mount on the bike is intact. This part of the molding, that has a bore to accept the connecting bolt, was cracked, but I joined the break with my soldering iron, and then melted black zip tie material to it as a means of lightly strengthening it. So that part of the unit should be good 'to go'.
An online search shows very little (nil) in the way of replacement for the 2023 LED. I emailed the local BMW bike outlet, and no reply all day yet. There are other mounting methods out there, but not like this system. Perhaps no BMW replacement exists yet outside of the production line.
I'm wondering if anyone here has gone the curiosity route and cut open the 2023 rear LED signal housing, to find it hollow, and where such hollow begins and ends. I was hoping maybe I could run wires along the outside, even in a 'trench' to conceal it. Pehaps you found a way to link to the wiring right behind the reflector. Whatever you may have found would be (pardon the expression) enlightening.