Real DS to emulated Pokemon trade / Mystery Gift?
While I maybe could try and search for another cartridge of that generation to save my pokemons, it would probably be faster to just emulate the game, and trade with that, with the additional benefit of backing up the pokemon themselves.
You're thinking backwards. That would not be faster or easier. You just a "pokemon box" type of storage for your Pokemon so you can transfer over to your newly created save on your existing cart.
- If you have a 3DS you could use PKSM
https://github.com/FlagBrew/PKSM/releases
- Then after that you can start a new game and transfer those pokemon back to HGSS.
Optionally, you could do what you originally suggested and use TWLSaveTool. Just because it's not maintained doesn't mean it doesn't work.
https://github.com/TuxSH/TWLSaveTool
Then copy the save over to your emulator.
NOTE: Wifi emulation does work on emulated DS (melonDS for windows). It's confirmed working with Pokemon. However, I do not believe you can connect to a real 3DS at this time. Read here for more information.
https://gbatemp.net/threads/after-several-years-ds-wifi-emulation-is-back.474947/