How can I build & maintain a personal, cross-platform 'toolkit' of commonly used applications?
For your own personal use, I think a GitHub repository with all of your bootstrap and installation scripts is a great idea. It serves as a portable way to download and distribute any scripts you need where firewalls and restrictions aren't an issue. 

As for within your organization, one suggestion that your IT might be open to as a compromise is to create a machine (barebones, VM, etc.) that is solely dedicated to downloading and storing the artifacts that you need. The criteria for this machine would be:


It has access to download artifacts (files, apps, etc.) from a source (preferably either the open internet or from white-listed addresses).
It is able to store the artifacts in a central location.
It is not able to access internal infrastructure
It is has auditing to track any artifacts that it downloads/stores


With this setup, you could download something like the latest version of Python onto the server (via logging in with credentials, API, etc.) and have it store Python in a central location for your other servers to download and install. You could store your artifacts in a multitude of ways (artifact repository, cloud storage, file server, etc.), but only that server should be able to upload to your storage. Your IT team could even set alerts when a new artifact is downloaded on this server or uploaded to your storage so that they can inspect it. 

I think this approach demonstrates a few things to your IT department:


These artifacts are important enough to you and your teams that you are willing to go out of your way to establish a safer and more traceable method to retrieve them.
You have given your IT department an automated way to track who and when an artifact was downloaded in the event an artifact is malicious and needs to be deleted and traced to a source (something they probably didn't have in place before).
The method you propose not only gives you more accessibility to the tools you need to be productive, but you are proposing ways to help your IT team be more productive by not having to manage all aspects of downloading artifacts.