About "doing my best":
"As much as" is part of the comparative structure, which means "as much as …".
"as" is a comparative conjunction here, which does not act as a sentence component, but only connects two comparison objects.
"I can" is actually an abbreviated form of "I can do", because "do" has been mentioned in the main clause, so it can be omitted in the clause. This omission is to avoid repetition.
On "Duo" as the object;
In this structure, "much" is a pronoun and refers to the number of an action or task mentioned above.
Much is the object of do, but because do is omitted in I can, much follows as directly.
Regarding whether "as" acts as a component:
In comparative adverbial clauses, "as" is a subordinate conjunction, not the subject, predicate or object of the sentence. Its function is to guide clauses and show that the relationship between clauses and main clauses is comparison.
On the attributive clause of "that";
The situation that you mentioned that "antecedents are usually led by that" does not apply in this sentence. Because "much" in this sentence is not the antecedent, and there is no "that" to guide the attributive clause.
To sum up, "try my best" in this sentence is a comparative adverbial clause, which is used to modify the verb phrase "try my best" in the main sentence. In this structure, "as" is a relative conjunction, "much" is the object of "do" (although "do" is omitted in the clause), and the whole clause means "try my best".