The deployment is really small; < 10MB. WEBSITE_WEBDEPLOY_USE_SCM was set to false. I didn't do this myself. Was set that way be default by the tooling when I stood up the App Service. I changed it to true. I then deleted and recreated the publishing profile and tried again. Now the operation completes successfully.
The weird thing is that I also deployed this exact same application code to a different App Service / App Service Plan that had the WEBSITE_WEBDEPLOY_USE_SCM value set to false and it works just fine. The only difference appears to be the publishing service associated to the target applications; the one that wouldn't work is associated with waws-prod-blu-167.publish.azurewebsites.windows.net and the one that worked without issue is associated with waws-prod-blu-137.publish.azurewebsites.windows.net.
Really strange and a huge pain to troubleshoot. Doesn't make any sense.