Where are the anti-antibodies created? In a mouse or in a human? Well, of course, you can't use humans to create antibodies so we are left with mice. So what's wrong with that?
Anti-antibodies will be created to the Fc (the constant section of the antibody) region of the antibody because the antibody is foreign and is treated as a foreign antigen.
So, you would have to use chimeric (or humanized) antibodies... I believe they are pretty expensive and takes a long time to make. Let's see that in current time. By now, I am sure you have heard about ZMapp, the drug that treats Ebola. Well, creating that drug is similar to this because they are creating antibodies as well. But I heard that a few doses of this drug takes 3-6 months to make! In this article , it mentions that it takes half a year to make ZMapp!
In conclusion, it's a real challenge, which is why we haven't found drugs that work like how you describe out in the market... But it's nice that you are thinking about this stuff...