Is the inversion in “Let’s see ʜᴏᴡ ᴄᴀɴ ᴡᴇ do this” an error for “Let’s see ʜᴏᴡ ᴡᴇ ᴄᴀɴ do this”?

You are right. The correct sentence would be

Now let's see how we can do this.

The incorrect form you've read demonstrates a fairly common English mistake among non-native speakers, especially those whose native language allows for omission of the subject pronoun (such as the Romance languages).

The confusion arises from three points:

• In a question, the word order would be "can we", whereas in a sentence or noun phrase, the order would be "we can", so the speaker/writer must remember which order to use in which case. It is an easy mistake to make as a non-native speaker.
• This noun phrase (the object of the verb "see"), includes the word "how", which is a question word. This can cause some confusion unless the speaker/writer takes a moment to consider that this is, in fact, not a question.

• In some languages, the subject pronoun "we" would not be used explicitly in either the question or statement form. Example (Spanish, no special characters):

  ..donde podemos encontrar el perro... ("where we can find the dog")
  ¿Donde podemos encontrar el perro? ("Where can we find the dog?")

  where the only difference is that one is written as a question.

Embedded questions normally don't allow Subject-Auxiliary inversion, though it's required for normal questions:

• How can we do this? (Aux can and Su we are inverted)
• *How we can do this? (No inversion; ungrammatical)

However, in spoken English (which is similar to the chatty, informal style used in many computer books), an embedded question complement can optionally invert the subject and the first auxiliary verb, like a regular question does.

• I wonder how we can do this. (No inversion; normal for embedded Q)
• I wonder how can we do this? (Su-Aux inverted; marked pragmatically)

There's a pragmatic distinction between the two forms, with and without the inversion. The presenting construction -- the one that resembles a real question -- is in fact intended to operate like a real question, and thus invite an answer from the addressee, rather than being intended to be interpreted as a simple rhetorical question.

For that reason, it's often encountered punctuated like a question; and sometimes a comma intonation -- or even a semicolon -- dashes in to separate the clauses.

• I wonder, how can we do this?
• I wonder -- how can we do this?

Now, this particular instance is in a piece of writing, where any question the author asks has to be rhetorical, after all, so this usage in this context probably is meant simply to increase the chumminess of the Boost.Interprocess salesman's pitch. If you're interested, there's an extensive literature on such syntactic marking in Pragmatics; the key term is "indirect question".

Executive Summary:

• I wouldn't worry about this (unless I were being paid to edit it for an ESL textbook).

The sentence should be written as

Now, let's see how we can do this.

Alternatively you can write

Now, let's see. How can we do this?