It really depends on your confidence level. It’s a glib answer but, if you’re asking, yes- you ought to use a pro.

Siting a bridge is another critical aspect. Apologies if this is stuff you already know, but just in case others look in…

Any fretted instrument only works if the scale length is right. The distance from the nut to the bridge is the scale length, and the 12th fret is halfway between. If not, the intonation flies out the window and the instrument is almost unuseable. Because strings are non-ideal (and frets are even less so), string saddles can be adjusted to compensate. Generally, you’ll see that the E,A,D saddles are closer to the back of the instrument than the GBE saddles.

Replacing a bridge means placing it in the right spot, because there’s only a certain amount of play with saddle-adjustment. A Bigsby trem is really adding to the complexity. I wouldn’t want to do it myself, but if you know what to do then there’s no law against it.

If you found the missing bit, you could stick it on with a cyanoacrylate glue (in England, it’s called “Superglue”) and it should stay in place. OTOH, you can get pre-cut nuts for popular guitars like the Fenders and Gibsons.

I agree with Lee, though. This job is quite critical and yet good guitar techs don’t charge much for it. I’m sure Georgia has quite a few, but post back if you draw a blank.

this can be a tricky job to do yourself, you need a new nut, they come uncut, and can be quite difficult to cut properly, you need some small files and a lot of patience, its not a difficult job to do, but it is a difficult job to get right, if it were my guitar i’d take it to the shop and let the pros do it.