It's actually quite smart, whether it was intentional or not, to switch from her real name of Lei/Leila to the fake name Ria, which is totally different. Since the slavers are a big organisation, and there could even be a kingdom noble behind them, allowing her real name to spread around could be quite dangerous. Outside of the barrier is the slavers' home turf, so even if good folks working for her family could potentially hear the name (if her real name was used all the time) and thus be able to save her, it's more likely the slavers or their associates would hear it first.
"Ria" isn't a fake name, it's a translation artifact based on the shortening of her real name, resulting from Japanese not having separate R and L sounds. That's why her family was using "Ria" in the last chapter's translation.
Her name is リーリア, pronounced somewhere between Rīria and Līlia. It represents a foreign name converted to Japanese orthography. I don't know if there is an official English version of her name, but I would probably go with something like Lyria. This makes the shortening to リア (Ria) more understandable.
Also, her late mother's name is クリア, which is pronounced Kuria or Kulia and probably intended to represent a name like Clea (KLEE-a). Ideally, the similarity with her mother's name (both ending with リア) would show up better in English, but I don't see any options that look good.
NB: Google Translate wants to render リーリア as Lilia even though it translates that back to リリア. So far, Leelia is the most natural version I can find that Google will render as リーリア, and it's not that good. Google thinks Lyria is pronounced the same as Luria (and thus renders it as ルリア), but that seems to be its error.