Elements with weak or blended lines are typically challenging to measure with the standard pipelines of spectroscopic surveys because these lines require a special treatment to measure precise chemical abundances. However, measuring the abundances of additional elements can open up opportunities to study a variety of topics from nuclear physics to Galactic chemical evolution. We present an analysis of high signal-to-noise spectra from the Apache Point Observatory Galactic Evolution Experiment (APOGEE) survey for ~120,000 red giant stars. To improve upon standard pipelines, we measure line-by-line abundances with the Brussels Automatic Code for Characterizing High accUracy Spectra (BACCHUS) and we employ a dedicated set of line quality flags as well as a prescription for identifying upper limits to provide precise abundance measurements. We use this methodology to create the BACCHUS Analysis of Weak Lines in APOGEE Spectra (BAWLAS) catalog, which provides abundances for Na, P, S, V, Cu, Ce, and Nd and the 12C/13C isotopic ratio for this sample of red giants, expanding the data available for stars observed by the APOGEE survey.