The longest I could find was
San Marino -> Romanians.
However, I am unsure of two things:
1) There is a possibility that the website I used to search for anagrams (Quinapalus Word Matcher) does not have complete results. I may have missed some equally valid and longer options.
2) Technically, "Romanians" does not have an entry in the current free online Merriam-Webster dictionary, but "Romanian" does. However, Romanians can be a valid plural form for Romanian, and typing Romanians into the dictionary redirects me to the entry for Romanian.
I'm not good at programming anyway, so I resorted to brute force (bad news: tedious; good news: I had time).
First, I copy-pasted part of the table in the site provided by OP. Next, I removed all the numbers and spaces, and then proceeded to sort them all by length, by hand. (Again, the downsides of not having a programmer's knowledge, but at least I wasn't breaking any rules set by OP! :P )
It was also fortunate that OP reminded me that only countries of length 8 letters or more are to be considered, so in the middle of sorting I removed the countries with less than 8 letters as well. Finally, I entered, one-by-one, each country name into the Quinapalus Word Matcher site.
At this point,
No anagrams of countries with length 10 letters or more appeared. After that, I saw Argentina -> "tanagrine", and San Marino -> "Romanians". Quick searches in Merriam-Webster (free online version) invalidated the first result.