Yes, I would indeed suggest Lviv (US State Department Level 3 which is the same as Egypt and most of Central America and Jamaica and parts of Mexico) and Montenegro (Level 1). Both I suspect safer than San Francisco (just my perception). If not to live, but just for tourism I also suggest they won’t disappoint. I know good guides in both.
I didn’t know a war was a "thing" especially when the "thing" was tied to the non-Schengen countries in general being at a higher risk. A quick look at a map and the Countries with land in Europe you will see that every country in Europe that borders Ukraine or Russia, except for Moldova is a Schengen country. With the one exception of Moldova there are no non-Schengen countries within 500 miles of the Level 4 part of Ukraine.
Interestingly enough is that while Schengen Countries like Italy, Germany, France (and a few others) are Level 2 by the US State Department, a great many of the non-Schengen countries are Level 1. The countries you will want to look at are: Albania, Azerbaijan, Bosnia and Herzegovina, Cyprus, Georgia, Ireland, Kazakhstan, North Macedonia, Montenegro, Serbia, Turkey, Ukraine and the UK.
While you were worried about the non-Schengen country risks with the war and while another was worried that Ukraine was in its entirety a war zone If you put the two ideas together you have to ask how safe it is to go to the Schengen Countries of Finland, Norway, Estonia, Latvia, Lithuania, Poland, Slovakia, Hungary, Romania as they all border either Russia (Level 4) or Ukraine (the Level 3 portion of the country). They are all Level 1.
MarkK, you are correct, the going rate in Budapest (US State Department Level 1) for about 700sf here is maybe $700 on average. Throw in another $125 for utilities and common costs. 700sf is about the median size here.
In Lviv and in Montenegro the prices will be a bit more. I know because I priced things about 2 years ago. So all a bit cheaper than Germany (Level 2 since we are making note of such things) and France (also Level 2).