What syntactic function does 'us' have here?

Solution 1:

The word us is the Direct Object of the verb meet in the Original Poster's sentence.

The syntactic functions (grammatical relations) in the sentence are:

Sentence level:

• Subject, We;
• Predicate, need you to meet us at the library at 7 tonight

Within the Predicate:

• Predicator (Head of the verb phrase), need
• Direct Object, you
• Catenative Complement, to meet us at the library at 7 tonight

Within the Catenative Complement:

• Marker of subordination, to
• Predicator (Head of the verb phrase), meet
• Direct Object, us
• (locative) Adjunct, at the library
• (temporal) Adjunct, at 7
• (temporal) Adjunct, tonight