After a few days of work, I am quite happy with the results. I can now create a brand new project in Codea, export it, and after only a few minutes, I receive an e-mail to install it on my device, with add-ons like music automatically integrated for me.
Even if you don't have a Mac you can leave turned on all the time, if you turn it on when needed, the build machine saves a lot of time building an exported project and installing it on your device.
As an added bonus, since everything, including add-ons, is in the Cloud, it is possible to modify or even add new add-ons directly on the iPad (I use Textastic, but any code/text editor supporting your cloud application would work).
I'm pretty sure I'm not the only one who will greatly appreciate having a Codea build machine, so I made everything available on GitHub.
https://github.com/jfperusse/CodeaProjectBuilder
If anyone tries it, let me know how that goes so I can improve the scripts and documentation!
You can also find my add-ons on GitHub, more to come soon!
https://github.com/jfperusse/CodeaAddons
Here's a video showing an example project with mp3 playback built completely on the iPad.