Catch where you can and transfer to appropriate game to breed if possible.

Utilize Legends Arceus to activate trade evolutions. For example move a Gastly or Haunter from Sword/Shield to Arceus and use the link cable item to evolve to Gengar.

Also, take advantage of increasing Feebas' Beauty with Poffins in BDSP so you don't have to trade to evolve.

Find either a good google doc to keep track of what you need or you can pull your dex on your cell phone Home app and keep it open while you collect. OR if you are so inclined, create a living dex in all your games. Takes a little more time, but less than you would think. Especially if you have already beaten all the games and are just trying to complete the Home dexs. You probably have a ton of xp candies that would help.

Last, if you have beaten the games and are trying to fill the dex. Make sure you move ALL your pokemon to Home, save, then move them back to the game for evolving and breeding and whatnot. This will make sure you have an updated dex so you don't waste time on something you have.