If it can't go to SWSH there's no reason for it to be able to go to LA/BDSP; it's all still using gen 8 Pokemon data.
It's because they store different movesets. Ever since the Home update for BDSP and Legends, most info is stored server-side for Pokemon rather than game-side, so stuff like Tera-type can be stored safely in the coding without it being reset. It's the same with stuff like TR rememberance and Dynamax level, BDSP and Legends don't keep track of that data at all, but Home does so when you move a Pokemon from SwSh to BDSP back to Sword it still can re-learn all of the TRs and it's dynamax level wouldn't be reset. G-max status is also kept track of but G-max Pikachu, Eevee, and Meowth can't be sent to BDSP or Legends to prevent getting Raichu, Persian, and Eeveelutions with a non-functional G-max status
Similarly, for Ribbons, Pokeballs, and Origin Markers introduced in BDSP and Arceus, they keep data lined up-coding wise within SwSh for any undefined ribbons and pokeballs without giving any updates to SwSh's coding at all after the home update happened. the Ribbons are invisible but still flagged, the origin mark just defaults of the Galar one allowing Pokemon caught in BDSP and Legends to obtain the Master Rank ribbon without needing to be Battle-ready marked while the pokeballs appear as normal Pokeballs but are still remembered in the codinglike how it was for Pokemon caught in Apriballs when traded to Diamond, Pearl, and Platinum worked.
I'm not saying they WILL allow Pokemon caught in SV to be sent to BDSP and Legends, but if they REALLY wanted to allow that it would be a lot easier to do than you think, for hyporhetical SwSh compatibility it mostly depends on whether or not SV stores seperate moveset data or not since I'm pretty sure they'd be too lazy to update SwSh since the Galar games didn't even get one for BDSP or Legends.