[Casiopea]

Both [a] and [b] are possible in both dialects (Sorry, Dippit):

[a] Let's write him. <short for: Let's write him a letter>
[b] Let's write to him. <short for: Let's write a letter to him>

That American speakers seem to prefer, as you've noted, the first structure to the second doesn't mean one is AmE and the other not. The difference you've come across has to do with efficiency. I am Canadian and I use both [a] and [b]:

[a] Let's write him. <short for: Let's write him a letter>
[b] Let's write to him. <short for: Let's write a letter to him>

The first structure has the pronoun "him" incorporated. That is, it's folded into the sentence, like this,

Let's write a letter to him => Lets write him a letter.

Incorporation changes the original order of the sentence and, in doing so, it makes the structure more efficient. For example, which structure below has less words, which one has more words?

Pat gave Sam a book. <incorporated>
Pat gave a book to Sam. <unincorporated>

The incorporated structure has less words; it's more efficient.

I use both structures.