Acinetobacter baumannii is a Gram-negative organism listed as an urgent threat pathogen by the World Health Organization (WHO). Carbapenem-resistant A. baumannii (CRAB), especially, present therapeutic challenges due to complex mechanisms of resistance to β-lactams. One of the most important mechanisms is the production of β-lactamase enzymes capable of hydrolyzing β-lactam antibiotics. Co-expression of multiple classes of β-lactamases is present in CRAB; therefore, the design and synthesis of “cross-class” inhibitors is an important strategy to preserve the efficacy of currently available antibiotics. To identify new, nonclassical β-lactamase inhibitors, we previously identified a sulfonamidomethaneboronic acid CR167 active against Acinetobacter-derived class C β-lactamases (ADC-7). The compound demonstrated affinity for ADC-7 with a Ki = 160 nM and proved to be able to decrease MIC values of ceftazidime and cefotaxime in different bacterial strains. Herein, we describe the activity of CR167 against other β-lactamases in A. baumannii: the cefepime-hydrolysing class C extended-spectrum β-lactamase (ESAC) ADC-33 and the carbapenem-hydrolyzing OXA-24/40 (class D). These investigations demonstrate CR167 as a valuable cross-class (C and D) inhibitor, and the paper describes our attempts to further improve its activity. Five chiral analogues of CR167 were rationally designed and synthesized. The structures of OXA-24/40 and ADC-33 in complex with CR167 and select chiral analogues were obtained. The structure activity relationships (SARs) are highlighted, offering insights into the main determinants for cross-class C/D inhibitors and impetus for novel drug design.