Turns out it was purely a network issue, that as of yet we still haven't actually figured out. But we tried hosting the app elsewhere and it was fine. That's my recommendation if anyone else has the same problem!
I've been following Microsoft's Teams C# tutorials found here, and have been successful for the most part. However, I cannot seem to get my app to work when I host the messaging endpoint myself rather than via their Azure service, which is not an option for me ultimately as the pricing is outrageous for what we need it to do.
I'm hosting the endpoint myself by publishing the sample project and ensuring it's externally available via HTTPS. I can access a custom tab within Teams, so I know that it's online, it's just the messaging endpoint that seems to fail with an "unable to reach app" error when I try and use the messaging extension via a chat window.
When debugging using dev tools, I get 502 error: Bot returned unsuccessful status code Forbidden, error code 1008. Every potential solution I've seen for similar issues hasn't worked for me thus far, though I still feel like it's something incredibly obvious. Are there special steps that need taking when hosting the endpoint yourself? The docs do a very lousy job of explaining the process, probably because Microsoft want you to pay to host the app on Azure.
Turns out it was purely a network issue, that as of yet we still haven't actually figured out. But we tried hosting the app elsewhere and it was fine. That's my recommendation if anyone else has the same problem!