I am sure this must be written somewhere but since I don't know a reference, let me sketch the computation. I hope there is no error in computation below.

As I commented in your related post, what you are dealing with is the sum of $t=k/2$ exponentials (let's assume $k$ is even, the case of $k$ odd is not really that different). I will also assume that $\log n<<t$ since this seems to be the regime you care about; the adaptation to the other cases I leave to you, I did not check the details.

Thus, $X_1=\sum_{i=1}^t E_i$, with $E_i$ exponential variables. Thus, for $w<1$, $P(X_1>t+w\sqrt{t})$ is of order $1$ while for $w>1$ but $w<<\sqrt{t}$, as long as $t\to\infty$, you have Gaussian tails: $$P(X_1>t+w\sqrt{t})\sim c_1 e^{-c_2 w^2}/g(t)\,,$$ for some function $g(t)$ that is essentially of polynomial growth and which I have not computed - it is $\sqrt{t}$ for moderately large $w$'s but later it may change. (For this you need to appeal to known results on precise moderate deviations for sums of independent variables, e.g. as in Petrov's book on sums of independent random variables; there is a big difference in your original question between $t$ fixed and $t$ growing in $n$ in that the tail estimate change from Gaussian to exponential, and this affects the scaling.). When $w$ becomes of order $\sqrt{t}$ or larger, then you get a different tail estimate, but I do not discuss this case in details - basically you get into large deviations regime and beyond.

Now, just follow the derivation of the law of the max: you get, for the range you care about, $$P(X_{\max}<t+z)\sim (1-c_1e^{-c_2 (z/\sqrt{t})^2}/g(t))^n\sim e^{-c_1n e^{-c_2 (z/\sqrt{t})^2}/g(t)}.$$ Now you choose $z=c_2'\sqrt{t\log n}-c_3\log\log g(t)\sqrt{1/\log n}+z'\sqrt{1/\log n}$. Then, with appropriate choice of constants, $$P(X_\max<t+z)\sim e^{-c_1e^{-c_4 z'}}\,,$$ which is Gumbel (note that the centering is $t+c_2'\sqrt{t\log n}-c_3 (\log\log g(t))/\sqrt{\log n}$). For this to work, I used that $\log n<<t$, so that $z/\sqrt{t}<<\sqrt{t}$.

I am sure this must be written somewhere but since I don't know a reference, let me sketch the computation. I hope there is no error in computation below.

As I commented in your related post, what you are dealing with is the sum of $t=k/2$ exponentials (let's assume $k$ is even, the case of $k$ odd is not really that different). I will also assume that $\log n<<t$ since this seems to be the regime you care about; the adaptation to the other cases I leave to you, I did not check the details.

Thus, $X_1=\sum_{i=1}^t E_i$, with $E_i$ exponential variables. Thus, for $w<1$, $P(X_1>t+w\sqrt{t})$ is of order $1$ while for $w>1$ but $w<<\sqrt{t}$, as long as $t\to\infty$, you have Gaussian tails: $$P(X_1>t+w\sqrt{t})\sim c_1 e^{-c_2 w^2}/g(t)\,,$$ for some function $g(t)$ that is essentially of polynomial growth and which I have not computed - it is $\sqrt{t}$ for moderately large $w$'s but later it may change. (For this you need to appeal to known results on precise moderate deviations for sums of independent variables, e.g. as in Petrov's book on sums of independent random variables; there is a big difference in your original question between $t$ fixed and $t$ growing in $n$ in that the tail estimate change from Gaussian to exponential, and this affects the scaling.). When $w$ becomes of order $\sqrt{t}$ or larger, then you get a different tail estimate, but I do not discuss this case in details - basically you get into large deviations regime and beyond.

Now, just follow the derivation of the law of the max: you get, for the range you care about, $$P(X_{\max}<t+z)\sim (1-c_1e^{-c_2 (z/\sqrt{t})^2}/g(t))^n\sim e^{-c_1n e^{-c_2 (z/\sqrt{t})^2}/g(t)}.$$ Now you choose $z=c_2'\sqrt{t\log n}-c_3\log g(t)\sqrt{t/\log n}+z'\sqrt{t/\log n}$. Then, with appropriate choice of constants, $$P(X_\max<t+z)\sim e^{-c_1e^{-c_4 z'}}\,,$$ which is Gumbel (note that the centering is $t+c_2'\sqrt{t\log n}-c_3 (\log g(t))\sqrt{t/\log n}$). For this to work, I used that $\log n<<t$, so that $z/\sqrt{t}<<\sqrt{t}$.