Suppose I have a full rank fat binary matrix $A \in \{0,1\}^{m\times n}$ with $m<n$. How can I find a matrix $A^{\dagger}$ such that $AA^{\dagger}=I_{m}$?
The traditional method used for matrices over $\Bbb R$, which yields
$$A^{\dagger} = A^T \left( A A^T \right)^{-1},$$
does not work in general because, even with full rank $A$, $AA^T$ may be not invertible. For example, if $A=\begin{bmatrix} 1 & 1 \end{bmatrix}$, then $A A^T = 1 + 1 = 0$, which is not invertible.
Using row operations and column operations, you can always make $A$ into the form $D=[I_m|O_{n-m}]$. In other words, there exist invertible $P$ and $Q$ of suitable sizes such that $A=PDQ$. Then $A^\dagger=Q^{-1}D^\top P^{-1}$. In fact, this works over any field.