I also play romanian subs on Plex client for Samsung SmartTV since 2013 without any issues (and a lot of other clients too).
Also, if the Plex server and clients are UTF-8 compliant and use UTF-8 exclusively, there is no actual (technical) meaning for a "character set / code page option". Love or hate Unicode :-) but UTF-8 can encode it all.
So, the first thing I would check: is my subtitle file UTF-8 encoded? Secondly: if the original (source) subtitle was ASCII -- did I properly convert it to UTF-8 (taking the character set / code page into consideration) or it was a "dirty" re-encoding to UTF-8 without properly taking into consideration the character set?
From my personal experience with romanian subs, I can tell you, unfortunately, almost all the subs you get from websites like titrari.ro / subs.ro / opensubtitles.org etc. are ASCII based encoded using ISO-8859-2 (or Windows 1250) character set. This is bad because: 1. the proper character set for Romanian ASCII encoding should actually be ISO-8859-16 not ISO-8859-2 and 2. ideally they should be UTF-8 encoded, not ASCII. So you need to take this into account when using such subs in Plex -- they should be converted to UTF-8 (taking into consideration the character set / code page of the source file), not just re-encoded as UTF-8.
If you're on a Linux based system, you could prepare a simple conversion script around: iconv -f ISO-8859-16 -t UTF-8 bad_ascii_sub.srt > good_utf8_sub.ro.srt and all your library would be updated in a matter of seconds. I guess on Windows you could do something similar (or use Notepad++).
Regarding the subtitle naming scheme, shocker-live advice is a good one too, see https://support.plex.tv/hc/en-us/articles/200471133-Adding-Local-Subtitles-to-Your-Media-- External Subtitle Files section for more details on this topic.