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As you have rightly pointed, the remote exam image is based on Fedora and we do not claim that it supports every device/hardware. If your hardware is incompatible, you will be required to use different hardware.
In your case, have you run the compatibility test prior to the exam? The compatibility tests have a section to detect audio and if your inbuilt mic is not supported, that test should have failed. (If it passes and the proctor could not hear you, it could simply be a case that the mic volume was low in the initial settings menu)
If you do not have compatible hardware, the mic issue can be fixed by using an external mic. (The easily available ones are web cameras or a basic headset with an inbuilt mic) and use it as the mic input in the settings menu.
As you were already asked to reschedule, please raise a ticket here, with details of your exam and the reason you were asked to reschedule.
Also as @Gineesh said, please try to run the compatibility tests multiple times a few days before the exam.