Beta-lactamase inhibitors are increasingly used to counteract antibiotic resistance mediated by beta-lactamase enzymes. These inhibitors compete with the beta-lactam antibiotic for the same binding site on the beta-lactamase, thus generating an evolutionary tradeoff: mutations that increase the enzyme's beta-lactamase activity tend to increase also its susceptibility to the inhibitor. Here, we investigate how common and accessible are mutants that escape this adaptive tradeoff. Screening a deep mutant library of the blaampC beta-lactamase gene of Escherichia coli, we identified mutations that allow growth at beta-lactam concentrations far exceeding those inhibiting growth of the wildtype strain, even in the presence of the enzyme inhibitor (avibactam). These escape mutations are rare and drug-specific, and some combinations of avibactam with beta-lactam drugs appear to prevent such escape phenotypes. Our results, showing differential adaptive potential of blaampC to combinations of avibactam and different beta-lactam antibiotics, suggest that it may be possible to identify treatments that are more resilient to evolution of resistance.