We don’t know
I haven’t seen any evidence that Ahsoka was aware of the significance of the name “Ashla.” We know that the Bendu used the word, but it is very old and undoubtedly remembers a great deal that the Jedi and Sith have forgotten. We know that Chava of the Lasat used the term, but we also know Ezra was unaware of its meaning, though Kanan might have known:
CHAVA: The Ashla led us to the pirate, then to you, so that we may find our
path to the new world.
EZRA: The Ashla?
CHAVA: The spirit of the galaxy.
EZRA: Sounds like the Force.
KANAN: The Force has many names, Ezra.
So far in canon, there’s not enough evidence to say whether the term “Ashla,” let alone its significance, was known to the Sith or the Jedi.
But more to the point, it wouldn’t matter if it were. Ashla was probably not an uncommon name, at least among the Togruta. There was at least one Togruta youngling named Ashla during the Clone Wars, whom Ahsoka might have known. Religious names are quite common in a variety of languages: Christian in English, Jesus in Spanish, and so forth. Could the Sith have known about this history? Sure. But it wouldn’t help them much. Would you consider the fact that someone was using the name “Christian” a strong piece of evidence? There are probably millions of Togruta named Ashla in the galaxy.
It seems most likely that Ahsoka simply chose that as her alias because it was a common name, or perhaps because she had known people named Ashla. If there’s any in-universe1 symbolism, chalk it up to the will of the Force.
1: Out-of-universe, it’s probably a reference to the fact that Ahsoka’s name was originally going to be Ashla.