"We broke his bottle."
Translation:שברנו לו את הבקבוק.
This sentence has more than one possible translation in hebrew, and this one is not the main. The simplest translation would be: "שברנו את הבקבוק שלו". A bit more grammatically complicated, though most used in spoken language, would look like: "שברנו את בקבוקו". One more possible translation would be: "שברנו את בקבוקו שלו", though this construction belongs to very high style and would not be used in everyday speech.
I disagree. The given translation is the most common. Sure it means approximately "we broke for him his bottle" or "we broke him his bottle", but this is the common way of expressing.
Your suggestion of "שברנו את בקבוקו" is certainly grammatical, but rare in common use. It is not more or less complex than "שברנו את הבקבוק שלו", as it just contracts the last two words, but most people prefer not to use the possessive suffixes.
As for "שברנו את בקבוקו שלו", this is not high style. It just emphasizes the bottle possession. Usually when some other bottles are also under discussion and we want to emphasize that it was his specific bottle.