Thank you - it seems the “handler” prompt sent me in the wrong direction. The issue was, when I initially modded the save I set the game to Scarlet out of curiosity, to see what effects it may have in-game. There were none; and so I forgot about the setting (and never changed it back).
I played the game fully, and so every Pokémon I caught had an origin game of “Violet” whereas my own player was set to “Scarlet”. I’m still unsure what the setting is supposed to do, but basically reverting back to Violet made my player’s HT ID match all the Pokémon again.
As for my issues with the level 7 raids, it seems I simply need to partake in a few raids level 5+ to actually unlock them. So my save is now totally usable in a legitimate way, without grinding for Tera Shards!