Nikon FE Mirror-Return Spring "BOINGGG" Sound
I think that is the sound of a spring in the mirror return mechanism. Some Nikons of that design do so as they age. The camera will continue to work normally. It has nothing to do with the shutter mechanism. It could be returned to quiet operation, provided that the required damper is still available. Such a “repair” would cost more than the camera is worth. At this remove, it would be difficult, if not impossible, to find a shop that would attempt to do so. If the camera is otherwise in good operating condition, I’d just use it as is.
I have several Nikon FE cameras. Some sound like that. Others are quieter. They all work as they should. A used camera that is at least 42 years old (made 1978 – 1983) can be expected to have changes to the non-metal parts: light seals, mirror stop pad, rubber dampers, O-rings, etc. Obviously, Nikon no longer produces replacement parts, and experienced and qualified technicians are mostly retired or dead now.
There are several rubber pieces associated with the mirror-up and mirror-down operations.
https://www.suaudeau.eu/memo/manuels_rep/reflexes/nikon/Nikon_FE_rm.pdf
You could inquire of a camera repair service about replacing the damper. My attitude is: “If it isn’t broken—don’t fix it. Your choice.