For this type of debugging it is easiest if you open a support ticket by emailing support@rstudio.com. Please include your deployment log, the version of RStudio Connect, the RStudio Connect configuration file, and any relevant logs.
The "RStudio Connect configuration file" is the file stored at /etc/rstudio-connect/rstudio-connect.gcfg, which is used to configure the entire RStudio Connect server.
It's also worth mentioning here that the preview release of the RStudio IDE (version 1.2.x) now has support for push-button publishing of Plumber APIs, which should help with some of the trickiness of using the rsconnect::deployXXX functions.