You cannot trade Pokemon between Shield and Brilliant Diamond....they're not compatible with one another. You can only trade with another copy of Shield or Sword in Sword and Shield and only with other copies of Shining Pearl or Brilliant Diamond with your copy of Brilliant Diamond.
For pokedex completion all you need is a Pokemon HOME box in at least one of the consoles and a nintendo account (you only need a subscription the get more than the first free box)
First transfer the pokemon from the first game to a HOME box...then from HOME to the other game. Much better than codes but trade evolutions won't work this way. Keep in mind to check which older pokemons in each game. Some moves were also changed between in the Sinnoh Remakes. But if you don't delete the moves while they are in Sinnohthey will still be back after sending them back to Sword & Shield.
Also remember to visit the Island of Armor Dojo to remove the Gigantamax ability (Red X symbol) from a Kanto pokemon you wish to transfer, Gigantamax is incompatible with Sinnoh games.