Every time I win a siege in the first Restoration of Erathia mission, the game closes out. This is the second time this happened, and I'd like to know what's going on

Maybe a missing file? There's always the same clip after this occasion.

If it's a clip, then maybe, like in Gothic1, scale video=0 within the config file could help; the game stopped crashing after that. Had the same within other titles. If there is such an option is worth a try.

Or if it happens to be a missing sound file, here's a list of what I could find from my old CD install https://imgur.com/a/evZUrEW

I see. Where can I find this config file?

Ppavee idea is absolutely good to test first.

then maybe try this
https://steamcommunity.com/app/297000/discussions/0/2656452469256192475/
different problem, solution might work

or
Get the HoMM III HD launcher mod.

or
https://steamcommunity.com/app/297000/discussions/0/4033602326804302170/

or
https://www.reddit.com/r/heroes3/comments/43mil0/heroes3_consistently_crashing/

Honestly, I don't know where and if there is a config file like in Baldurs Gate and so on.

But I remember that in Kingpin there was a problematic intro, and deleting that intro helped to start a game. Maybe deleting the whole clip will help, too.

For more options, more ideas are needed. Good luck!