Representing a Linear Transformation as a Matrix

Your issue is that you're still thinking of the $2\times 2$ matrices as matrices. If you're going to have them be acted upon by transformations, and you need those transformations to be represented by matrices, you need to represent the elements (some $2\times 2$ matrices) of the base vector space (the space of $2\times 2$ matrices) as column vectors.

This may seem confusing because it's no longer clear what space each matrix or vector is associate with. You could attack this problem by adding some notation to make it clear.

Let the vector space that the $2\times 2$ matrices act upon be denoted $V$. Then let $M^{2\times 2}$ denote the set of $2\times 2$ matrices associated with linear transformations on $V$. Then what you have could be expressed like this:

$$\begin{bmatrix} a & b \\ c & d \end{bmatrix}_V \equiv \begin{bmatrix} a \\ b \\ c \\ d \end{bmatrix}_{M^{2\times 2}}$$

The subscript says what vector space each matrix or column vector is associated with.

Then your answer is merely

$$\begin{bmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}_{M^{2\times 2}}$$

which is correct.

This approach of using subscripts to denote what vector space the matrix or column vector is associated with is by no means standard. I do think this is one reason why higher-level treatments of linear algebra sometimes don't even use matrices: when you need to talk about a set of matrices itself as a vector space, things get a little confusing.