For $T \in L(E,F)$ continuous surjective linear operator between Banach spaces $E$ and $F$ we have that :
$Ker(T) $ admits a closed complement $L$ in $E \implies T$ admits a continuous right inverse
I can construct the linear right inverse $S y= Proj_L x $ s.t. $Tx = y $ and prove it is well defined (T is surjective and Tx = Tx' => Proj x = Proj x') and linear.
However, I am not sure on how to prove $S$ is continuous.
I think I can assert that the restriction of $T$ to $L$ is bijective from $L$ to $F$ so that $S=T^{-1}$ from $F$ to $L$ is continuous by the open application theorem, and extends $S$ to have $E$ as a codomain.
Is there any cleaner approach or explanation to this ?
You have shown that the problem can be reduced to the case $N(T) = 0$. In this case, the fact that $T^{-1}$ is continuous is not very different from the open mapping theorem. (See this.) So there's not likely to be a significantly simpler way.
Take the inverse of the restriction of $T$ to $L$.