Do you have the vocal isolated? If so then you can add distortion and a loudspeaker effect by playing that through a speaker cranked up and re-recording. That’s the old fashioned way and not really easy in FCP.
Second is to take it to an audio facility that has the ability to pass it through various plugins, starting with a convolution reverb for the loudspeaker effect and a distortion plugin.
Either way, it is beyond the scope of FCP. If soundtrack pro supports VST plugins, then you might be able to do it that way. I never use STP so I can’t advise beyond that. Given that a music video is usually for broadcast, I would take the job to a sound post pro who can do all that you want and make sure the end product is broadcast legal.