Prove that $(\mathbf{AB})^{T} = \mathbf B^{T}\mathbf A^{T}$ where $\mathbf A$ and $\mathbf B$ are matrices

It's incorrect, you cannot transpose a specific entry of a matrix (unless you treat it as 1x1 matrix, but that's not going to get you anywhere).

To do it correctly you need to write $$ \big(({\bf AB})^T\big)_{i,j} = ({\bf AB})_{j,i}$$ and later after using the formula for the entries of product of matrices, you'll go back with $$ {\bf A}_{j,k} {\bf B}_{k,i} = ({\bf A}^T)_{k,j} ({\bf B}^T)_{i,k} = ({\bf B}^T)_{i,k} ({\bf A}^T)_{k,j}$$