The Chandra X-ray observatory has discovered several dozen anomalously X-ray-bright jets associated with powerful quasars. A popular explanation for the X-ray flux from the knots in these jets is that relativistic synchrotron-emitting electrons inverse-Compton scatter Cosmic Microwave Background (CMB) photons to X-ray energies (the IC/CMB model). This model predicts a high gamma-ray flux which should be detectable by the Fermi Large Area Telescope (LAT) for many sources. GeV-band upper limits from Fermi/LAT for the well-known anomalous X-ray jet in PKS 0637-752 were previously shown in Meyer et al., (2015) to violate the predictions of the IC/CMB model. Previously, measurements of the jet synchrotron spectrum, important for accurately predicting the gamma-ray flux level, were lacking between radio and infrared wavelengths. Here we present new Atacama Large Millimeter/submillimeter Array (ALMA) observations of the large-scale jet at 100, 233, and 319 GHz which further constrain the synchrotron spectrum, supporting the previously published empirical model. We also present updated limits from the Fermi/LAT using the new `Pass 8 calibration and approximately 30% more time on source. With these deeper limits we rule out the IC/CMB model at the 8.7 sigma level. Finally, we demonstrate that complete knowledge of the synchrotron SED is critical in evaluating the IC/CMB model.