Essentially, we're working in a version of the quantum plane, with the scalar $q$ replaced by the variable $z$. This is a Noetherian domain, so we can invert the set of non-zero elements on either the left or the right to obtain a division ring, and these two rings (left/right fractions) are equal.

Rephrased in these terms, you're asking if and how the right fraction
\[\Big(B_0(y,z) + B_1(y,z)x + x^2\Big)\Big(B_0(y,z)z^{2n} + B_1(y,z)z^nx + x^2\Big)^{-1}\]
can be rewritten as a left fraction $g(x,y,z)^{-1}f(x,y,z)$.

This is definitely always possible, for any $n$, since the ring of left fractions and the ring of right fractions are equal.

To find an expression for $f$ and $g$, you can use linear recurrence relations. Let $k(y,z)$ be the fraction field of the commutative polynomial ring $k[y,z]$, and $\alpha: k(y,z) \rightarrow k(y,z): y \mapsto zy, z \mapsto z$.

Define the Laurent power series ring $k(y,z)((x;\alpha))$ by
\[k(y,z)((x;\alpha)) = \{ \sum_{i \geq n} a_i x^i : a_i \in k(y,z), n \in \mathbb{Z}, x a_i = \alpha(a_i)x\} \]
Our ring of fractions $k(y,z)(x;\alpha)$ embeds into this larger division ring, and now you can easily generalise to this case the commutative result which says that a power series represents a fraction if and only if it satisfies a linear recurrence relation. I used this set of notes, which I found quite clear and easy to follow.

Keeping careful track of what side your fractions are on, you should be able to obtain results on recurrence relations for both left and right fractions. This should give you the computational tools to translate the right fraction above into a left one, for any polynomials $B_0$, $B_1$, $B_2$ and any $n$.

Note: technically the $f,g$ you obtain will be elements of $k(y,z)[x;\alpha]$. However, it is now easy to multiply through by the common denominator of the coefficients (which are elements of $k(y,z)$) to obtain $f,g \in k[y,z][x;\alpha]$.

Editing to add more detail to address comment below.

Based on the notes I linked to above, we find that a power series $\sum_{i \geq 0} a_i x^i$ represents a left fraction $g^{-1}f$ if and only if there exists some integer $n$ and fixed coefficients $c_1, \dots, c_n \in k(y,z)$ such that
\[a_{i+n} = c_1 \alpha(a_{i+n-1}) + c_2 \alpha^2(a_{i+n-2}) + \dots + c_n \alpha^n(a_i)\]
for all $i \geq 0$. In this case, $g$ has the form $1-\sum_{i=1}^n c_i x^i$ and $f$ has degree $\leq n-1$. (Note that some of the $c_i$ are allowed to be zero.)

The element $(1-xy)^{-1}$ mentioned below has the form
\[1 + zyx + z^3y^2x^2 + z^6y^3x^3 + \dots = \sum_{i \geq 0} z^{\frac{i(i+1)}{2}}y^ix^i\]
as a power series so its coefficients satisfy a linear recurrence relation of the form $a_{i+1} = zy\alpha(a_i)$. (Compare this to the statement above about the relation between the denominator of a fraction and its recurrence relation constants.)

Meanwhile, for a *right* fraction, it turns out to be easier to write the coefficients of the power series on the right. Now a power series $\sum_{i \geq 0}x^ib_i$ represents a right fraction $fg^{-1}$ if and only if there exists an integer $m$ and constants $d_1, \dots, d_m$ such that for all $i\geq 0$, the coefficients $b_i$ satisfy
\[b_{i+m} = \alpha^{-1}(b_{i+m-1})d_1 + \dots + \alpha^{-n}(b_i)d_m.\]

I'm afraid I don't know if you'll end up with a modified version of the fraction you suggest: I don't have time to work through the details right now. Based on the techniques in the notes linked above, you should be able to write your right fraction quite easily as a power series with coefficients on the right (hint: there will only be three recurrence relation constants, and the last one will be zero), then switch coefficients over to the left and look for a recurrence relation on the left - I suspect this will be the tricky part. On the plus side, if it works you'll have a nice explanation as to *why* it works!