I'm wondering if this SOS response is a response to antibiotic treatment or to DNA damage. I suspect the latter.
If a bacterium has sustained a large amount of DNA damage (from whatever source), it would be helpful to activate a set of DNA polymerases that aren't so picky about error checking. A last resort if you will. A lot of these bacteria will probably die due to serious damage to important genes, but a few may get though, with an extremely high mutation rate. So the development of antibiotic resistance may just be a side effect of a simple survival mechanism rather than the whole purpose behind it.
A good test would be to see if this mechanism is activated by antibiotics that don't result in DNA damage.
However it isn't unusual that bacteria have evolved various defences against antibiotics. After all, microorganisms have been fighting amongst themselves for billions of years, and antibiotics are just some of the weapons they have been using. For example, a bacterium I worked on many years ago (Klebsiella planticola) naturally carried a beta-lactamase enzyme, which sole purpose is to defend against penicillin. Presumably it was exposed to penicillin in the wild, not from human intervention.
Marty