This is as good a summary as any...
An interjection is a word expressing some kind of emotion. It can be used as filler. Interjections do not have a grammatical function in the sentence and are not related to the other parts of the sentence.
If an interjection is omitted, the sentence still makes sense. It can stand alone. (italics mine).
I suggest that in OP's example, "yes" is an interjection. So if we remove it, we're left with...
"I'm in South Ossetia [?] I managed to get a visa!"
...where I've left [?] in there because it's not obvious which of the original punctuation marks were part of the interjection. You'll note from the link above that in all their examples, the interjection is a capitalised word followed by an exclamation mark.
I see no reason for OP to deviate from that pattern. The original should have been...
"I'm in South Ossetia. Yes! I managed to get a visa!"
...in which case there would be no problem in removing the interjection...
"I'm in South Ossetia. I managed to get a visa!"
In principle, the period in that final version could perhaps be a dash. But in practice hardly anyone would write "Yes! - I did it!", so if we're going to be consistent, "chatty" dashes aren't really helpful.
TL;DR: Don't use dashes in the first place. Stick to the standard orthography. Capitalise the first letter of the interjection, and follow it by an exclamation mark (or period). So you could remove it if you wanted to.