Step 1: Reframe by resistance threat. Catheter UTI plus pneumonia in a bedridden patient screams nosocomial flora that frequently manufacture beta-lactamase enzymes capable of inactivating ordinary penicillins and cephalosporins.
Step 2: Decide the counter-strategy. The pharmacological fix is to pair the beta-lactam with a beta-lactamase inhibitor, which neutralises the bacterial enzyme and restores the killing power of the antibiotic.
Step 3: Test each option. Amoxicillin alone is vulnerable; second and third generation cephalosporins alone give no enzyme protection. Only the inhibitor-combined beta-lactam directly addresses the resistance mechanism.
Step 4: Hence the empirical pick that covers the organisms causing both the UTI and the pneumonia is the beta-lactam plus beta-lactamase inhibitor.
\[\boxed{\text{Beta-lactam with beta-lactamase inhibitor}}\]