When I was under the age of 3, I used to pray for the Devil, that he might too be saved.
I told my parents this and was badly scolded, but continued to pray for the Baphometic one.

"Does the LBRP and LBRH banish the Qliphoth? Or is that an altogether other ritual?"

Broadly, yes.

Now, if you mean "clear them from my system," then that's a bigger thing that this only addresses over time. (Consider it similar to the question, "Would doing a pentagram ritual remove a long-standing bad habit?" The answer would be, "Well, not just one pentagram ritual...")

But in the sense of clearing the immediate space for an immediate, short-term purpose, yes; and in the sense of gradually strengthening your purity and fortification over time, yes.

If it's as strong as a "haunting" requiring a full exorcism - a very serious infestation of malignant, atavistic, soulless presence in a particular place - the exorcism would be a separate ritual.

"I remember reading about another more obscure way of doing this, is this necessary for those
who are Neophyte or Zelator grade or is this something else entirely. I remember reading
that a Magus sometimes works with "healing" the Qliphoth but I figured this is much different than
banishing them."

That's an entirely other area unrelated to your original question above.