Thinking about my comment again, maybe my remark on necessity of using left multiplication with $\Gamma$ was too hasty.
First, some general comment on the coset change in the specified situation.
Assume that there is a matrix $A\in KZ$ such that
$$
\left(\begin{array}{cc}
t^3&bt^2+t^{-1}+lt\\0&1
\end{array}\right)=
\left(\begin{array}{cc}
t^3&\frac{y-m}{x-l}\\0&1
\end{array}\right)\cdot A
$$
Then, working out the matrix multiplication, it is clear that
$A=e_{12}(f)$ with
$$
ft^3=bt^2+t^{-1}+lt-\frac{y-m}{x-l}, f\in \Theta_\infty.
$$
Rephrasing, the right coset representatives can be changed if and only if
$bt^2+t^{-1}+lt$ and $\frac{y-m}{x-l}$ agree up to
$t^3\Theta_\infty$. The right cosets are the same if and only if
Takahashi's Proposition 1 can be applied to prove it.
However (and contrary to the formulation of the question), there are situations where that is actually possible (I think): let's look at the simplest example, which is $l=0$. In this case, we can start the Laurent series expansion at infinity: $\frac{y-m}{x}=t^{-1}-\frac{m}{x}$. As $x^{-1}=at^2$ with $a\neq 0$ up to terms of order $\geq 3$, we choose $m=-ba^{-1}$ and find that $$ \frac{y-m}{x}=t^{-1}+bt^2+ft^3 $$ for some $f\in\Theta_\infty$. In this case, it is in fact possible to change the right coset representative as required in Takahashi's paper.
[Added later:] In the general case, we make the following Laurent series expansion (correcting some mistakes in the reformulated question): $$ \frac{y-m}{x-l}=\frac{m-y}{l}\cdot\frac{1}{1-\frac{x}{l}}= \frac{m-y}{l}\left(-\sum_{n=1}^\infty\frac{1}{\left(\frac{x}{l}\right)^n}\right) =\frac{y-m}{l}\left(\sum_{n=1}^\infty\frac{l^n}{x^n}\right)=\left(t^{-1}-\frac{m}{x}\right)\sum_{n=0}^\infty\frac{l^n}{x^n}.$$
Now we write $x^{-1}=a_2t^2+a_3t^3+T_{geq 4}$, and expand the above product: $$ \left(t^{-1}-\frac{m}{x}\right)\sum_{n=0}^\infty\frac{l^n}{x^n}=\left(t^{-1}-\frac{m}{x}\right)\left(1+l\left(a_2t^2+a_3t^3+T_{\geq 4}\right)+R_{\geq 4}\right)=\left(t^{-1}+la_2t+la_3t^2+S'_{\geq 3}\right)-m\left(a_2t^2+S''_{\geq 3}\right) $$ with all the $T_{\geq 4}, R_{\geq 4},S'_{\geq 3},S''_{\geq 3}$ power series of the respective valuation. We collect the terms of order $\leq 2$ to get the following equality modulo $t^3\Theta_\infty$: $$ \frac{y-m}{x-l}=t^{-1}+la_2t+\left(la_3-ma_2\right)t^2. $$ I guess it is not a problem to choose $x$ such that $a_2=1$, and then the above allows to set $m=la_3-b$. This should show that also in the case $l\neq 0$, it is always possible to change the coset representative as required in Takahashi's paper.
Maybe the argument Takahashi had in mind was to see the case $l=0$ and then apply Proposition 6 of the paper to reduce the general case to the special case $l=0$.