I doubt it's a bandwidth issue. Other than downloading the client files there's nothing to really do on client load. The strange thing is that it's the same users and therefore the same computers/browsers that have an issue connecting to your server, but they can seemingly connect to our demo server fine. I was thinking something like blocking cookies or popups or something like that.
I've never used LTI with BBB or edX so I'm not very familiar with it, but maybe the issue is with how the resource is loading. It looks like there's a configuration option for "launch target", are you using the "new window" option? I see in the image in the first post that there's the text "LTI CONSUMER (EXTERNAL RESOURCE)" above the client. I think that might mean that the client page is being loaded in an iframe instead of directly and might be getting tripped up by some sort of browser security.