Computing classical pushforward via Quotient Stacks

$\require{AMScd}$I am in the process of understanding how to work with (quotient) stacks. In my case, it is usually helpful to get my hands dirty so I like to come up with examples where maybe using stacks can make an argument more transparent. I recalled an exercise that I did when preparing for one of my algebraic geometry courses where they asked you to compute $f_*\mathcal{O}_{\mathbb{P}^1}$ where $f \colon \mathbb{P}^1 \to \mathbb{P}^1$ is given by $[x_0:x_1] \mapsto [x_0^2:x_1^2]$ (I will not indicate the ground field, if it makes it easier let us assume that all schemes are over $\mathbb{C}$). If I am not mistaken, one can show that $f_*\mathcal{O}_{\mathbb{P}^1} \cong \mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(-1)$ and the way I would have done it then was using local coordinates and computing the glueing isomorphism in the intersection.

I would like to do this without going through this local computation but rather see if one can come up with an alternative argument using quotient stacks. My intuition is that somehow one could use the morphism between classifying spaces $$ \lambda \colon B \mathbb{G}_m \to B \mathbb{G}_m $$ induced by the character $\chi \colon \mathbb{G}_m \to \mathbb{G}_m$ sending $t \mapsto t^2$. Then one can also see $\mathbb{P}^1$ as the semistable locus of the quotient stack $\mathfrak{X} = [\mathbb{A}^2/\mathbb{G}_m]$ where $\mathbb{G}_m$ acts with weights $1$ in all variables and the linearisation of the $\mathbb{G}_m$-action given by twisting the trivial bundle by the character $t \mapsto t$. In other words, $$ \mathbb{P}^1 \cong \left [ \frac{(\mathbb{A}^2 \setminus 0)}{\mathbb{G}_m} \right ] $$ My intuition is that now one could consider the Cartesian square (of stacks):
\begin{CD} X @> >> B\mathbb{G}_m \\ @VVV @VVV \\ \mathbb{P}^1 @> >> B\mathbb{G}_m \end{CD} Then I would like to factor $f \colon \mathbb{P}^1 \to \mathbb{P}^1$ through $X$ by using some universal property of the fiber product (or getting some better description of what $X$ is) and finally conclude with a cohomology and base change/flat base change argument (here I guess there is some repreresentation theory involved). To be honest, I am still struggling with these kind of computations/arguments, so I would appreciate any partial or complete solutions, as well as any comment indicating whether if my idea is even feasable.