> The alternative being building and deploying a new image, which would likely take significantly longer

You said the image was Python, though? Using that is way easier and faster. https://news.ycombinator.com/item?id=48558763

If all you need to know is that it can connect:

python3 -c 'import socket as s;s.create_connection(("8.8.8.8",53))'

or http:

python3 -c 'from urllib.request import*;print(urlopen("http://example.com").status)'