How do I name a binding antibody? “Binding antibody to”, “binding antibody against”, or “anti-[antigen] binding antibody”?
The adjective “binding” is not used to qualify “antibody” in English scientific usage because linguistically it is thought to belong with the antigen the antibody binds. 

Generally it is omitted if the prefix “anti” is used, e.g. 


  anti-cetuximab antibody


(e.g. “two or more anti-hapten antibodies” here)

However if you wish to refer to the particular small portion of the antigen that is recognized (the epitope), you might write, e.g. 


  tyrosine-binding antibody


(e.g. “Hapten‐binding antibodies…” here)

As far as the use of prepositions is concerned, “to” is used with binding, although it is optional. However generally with the antigen, e g. 


  cetuximab binds to the antibody 


(e.g. “Antigen binds to the antibody on the B-cell membrane.” here)

But


  the antibody binds cetuximab


(e.g. “An additional VNAR, Help6, which binds Hepatitis B precore protein” here.)

Completely illogical, I know, and this distinction is not universal. 

However, “against” is used in relation to preparing or raising antibodies, e.g.


  We raised an antibody against cetuximab…


(See the legend to Fig 1. in this paper.)

So in general one would write:


  
  Assay of anti-cetuximab antibodies in serum
  


Addendum: Binding antibodies v. Neutralizing antibodies

The revised question makes it clear that the authors are referring to “binding antibodies” as a specific contrast to “neutralizing antibodies”. This expression was not known to me (I am not an immunologist), and, although the poster documents it, its use would appear to be limited, perhaps because the topic is or was controversial.

The question remains, how to express this idea in a clear and unambiguous manner, with the specific problem that the poster is a translator, not an editor. I would still try to avoid the use of the word “binding” in the expression. I have two suggestions:


I believe that an alternative expression would be “non-neutralizing antibody” (the paper just quoted does not hyphenate this term, but others do, and I would). If this belief is correct, one could write:



  
  Assay of non-neutralizing anti-cetuximab antibodies in serum
  



Alternatively one could use the abbreviation, “BAb”, from the paper cited by the poster, although this would have to be defined somewhere.



  
  Assay of anti-cetuximab BAbs in serum