Thanks, Ryan. A difficult question indeed! I first want to make sure I understand your question, so I will unpack it. I apologise for the length of my answer. I think this is needed to be understandable and clear.
I assume you know that DNA codes for polypeptide chains and antibodies contain two different such chains, a light and a heavy chain. I assume you had in mind that “your artificial DNA” contained some genes to code for antibody polypeptide chains? However, a small amount of artificial DNA will only “code for” a few polypeptide chains. So, what did the artificial DNA you had in mind code for? I imagine a few polypeptide chains for a few antibody molecules. A cell given such DNA could in principle only make a few different antibody molecules. This could not provide protection against many invaders.
Your question mirrors one faced by immunologists in the mid/late 1950s. Antibodies were known to be highly specific in their interaction with antigen and they could be raised against any foreign antigen. There must therefore be a gigantic number of different antibodies and their corresponding genes. What was the origin of these genes?
I describe one visionary insight, in order to cut to the core of what we now know is true. Most genes are inherited intact from parents to offspring. Joshua Lederberg, the bacterial geneticist, suggested something radical in a 1959 Science article. He proposed that parents provide their offspring with assembly kits to assemble genes coding for immunoglobulin light and heavy chains. This process is outlined in my Spiral Immunology, linked to this site. The assembly process involves joining bits of DNA together in a way that involves some randomness to potentially create genes coding for functional antibody chains. Because the process has random steps, the antibodies coded for can have specificity for foreign or self-antigens or for both. Thus, this method of generating genes must be coupled with a mechanism of self-nonself discrimination to prevent/minimize autoimmunity.