This is much simpler than you make it. First, as gamma (also exponential) distributions have different parametrizations in use, you need to specify which you mean. From your formulas you seem to assume that $Y_i$ has exponential distributions with scale parameter $\beta$, and then the sum $S$ oof $n$ independent copies is $\mathcal{Gamma}(n, \beta)$, using the shape/scale parametrization https://en.wikipedia.org/wiki/Gamma_distribution.

The transformation to $U$ is now simply a multiplication with $2/\beta$, and this simply multiplies the scale parameter with the same constant, giving that $$ U=\frac{2S}{\beta} \sim \mathcal{Gamma}(n, 2) $$ There is no need for your use of Jacobians!

This is much simpler than you make it. First, as gamma (also exponential) distributions have different parametrizations in use, you need to specify which you mean. From your formulas, you seem to assume that $Y_i$ has exponential distributions with scale parameter $\beta$, and then the sum $S$ of $n$ independent copies is $\mathcal{Gamma}(n, \beta)$, using the shape/scale parametrization.

The transformation to $U$ is now simply a multiplication with $2/\beta$, and this simply multiplies the scale parameter with the same constant, giving that $$ U=\frac{2S}{\beta} \sim \mathcal{Gamma}(n, 2) .$$ There is no need for your use of Jacobians!