Should you say, "Forgive us our trespasses, as we forgive those who trespass against us" or "as we forgive them who trespass against us"?

Say it whichever way you like.

The pronouns those and them are both correctly used as objects in this passage.

Whom did Robert Herrick urge to go a-Maying? Robert Herrick urged Corinna, in "Corinna's Going A-Maying" (1648).
What is the last line of Little Caesar (1930)? The last line of Little Caesar (1930) was "Mother of Mercy, is this the end of Rico?"
Who said, “From what I hear about Communism, I don’t like it because it isn’t on the level”? Gary Cooper said, "From what I hear about Communism, I don't like it because it isn't on the level".
What did John Hancock (1737-93) do to earn his place as president of the Continental Congress? John Hancock's (1737-93) most distinctive contribution to the rebel cause was money which earned him a place as president of the Continental Congress. Hancock was a merchant who had inherited a fortune from…
How long did the Berlin Airlift last in 1948? The Berlin Airlift lasted for ten months in 1948 and 1949. 1,000 planes of the Western powers flew food, fuel, and other necessities to the two million civilians in West Berlin, then under…
What did Thomas Edison use as a filament in the first successful electric light bulb? After experimenting with thousands of materials, Thomas Edison discovered in 1879 that a scorched cotton thread, the equivalent of a carbon wire, was the filament he needed. It was one that would glow…