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. 

I am not entirely certain about the general case, because I did not do the Laurent series expansion.