Hello Duy,
It is an interesting problem, I was keen to understand why its happening. I searched on google and found the following:
IE & Chrome use the Windows OS root certificate store as a valid repository to build the certificate chain and hence when you add a new root certificate via distribution, they are automatically accessible and hence it would work.
FireFox doesn't trust the OS root certificate store, as FireFox assumes that any user with sysadmin privileges may be able to update the Windows root certificate store. So if your system was compromised, a hacker may add a fake root certificate into your OS root certificate store and trick you into believing that the site you are accessing was a legitimate one.
This is documented in the Mozilla thread (scroll to the bottom half).
https://bugzilla.mozilla.org/show_bug.cgi?id=432802
Although it was reported as a bug, the security admins / members / developers of the mozilla project say that this won't be added to mozilla in the future too.
So unfortunately, it looks like export of root certificate store from IE and import into Mozilla maybe the only option, or the other option that you suggested.
Regards,
Siddhesh