Local GCH absorbs sums and squares

Fix $a\in X$. The two copies of $X$ inject into $\mathcal{P}(X\bigsqcup \{\ast \})$ by $x\mapsto \{x\}$ and $x\mapsto \{x,\ast \}$. A bijection $X\bigsqcup 1\approx X$ transports this to $X\bigsqcup X⪯\mathcal{P}(X)$. Also $X⪯X\bigsqcup X$.

Local GCH makes $X\bigsqcup X$ equinumerous either with $X$ or with $\mathcal{P}(X)$. The latter would inject the power set into finite sequences: use $(x)$ for the first copy and $(a,x)$ for the second. This is impossible. Thus $X\bigsqcup X\approx X$.

The map taking a subset of a tagged disjoint union to its two component subsets is a bijection $\mathcal{P}(X\bigsqcup X)\to \mathcal{P}(X{)}^2$. Transport along the previous bijection gives $\mathcal{P}(X{)}^2\approx \mathcal{P}(X)$. Singleton coordinates inject $X^2$ into this product, while $x\mapsto (x,a)$ injects $X$ into $X^2$.

Apply local GCH to $X⪯X^2⪯\mathcal{P}(X)$. The power-set endpoint would inject $\mathcal{P}(X)$ into length-two sequences, again impossible; the remaining endpoint is $X^2\approx X$. Together with the product-of-powersets bijection this proves all assertions.