Coding a well-ordering of $2^{|\gamma|^{\lt \kappa}}$ in $P(P(P_\kappa\gamma))$ while preserving relative constructible hierarchy The title might be a little bit misleading because it is my main question but the setting is in large cardinal theory. But I had no better idea.
So to introduce some notation, for any normal ultrafilter $W$ over some $P_\kappa\gamma$, let $j_W: V \prec M_W$ be the respective ultrapower embedding followed by a transitive collapse. Now this is the exercise I have trouble with:

Exercise. If $\kappa \le \gamma$ and $\kappa$ is $2^{|\gamma|^{\lt \kappa}}$-supercompact, then
  $$(2^{|\gamma|^{\lt \kappa}})^+ = \sup(\{j_W(\kappa): W \text{ is a normal ultrafilter over } P_\kappa\gamma\}).$$
  Hint. One direction is clear. Suppose $\xi \lt (2^{|\gamma|^{\lt \kappa}})^+$. Let $A \in P(P(P_\kappa\gamma))$ be such that $\xi \lt (2^{|\gamma|^{\lt \kappa}})^{+L[A]}$ and, by a previous theorem, $W$ a normal ultrafilter over $P_\kappa\gamma$ with $A \in M_W$. Show that $\xi \lt j_W(\kappa)$.

The part that is a little bit confusing for me, is to find the $A$ in the hint.
So what I have done thus far is this: 
Let $R \in P(2^{|\gamma|^{\lt \kappa}} \times 2^{|\gamma|^{\lt \kappa}})$ be a well-ordering of $2^{|\gamma|^{\lt \kappa}}$ of order-type $\ge \xi$. And let B be the image of $R$ under the Godel pairing function, so that $B \in P(2^{|\gamma|^{\lt \kappa}})$. By definability and the fact that $B$ is a set of ordinals we have that $\xi \lt (2^{|\gamma|^{\lt \kappa}})^{+L[B]}$. I am stuck here.
There are two possibilities which come to my mind for completing this argument:
$(1)$ Using choice we identify $B \in P(2^{|\gamma|^{\lt \kappa}})$ with some $A \in P(P(P_\kappa\gamma))$. Then using $A \in M_W$ and some closure properties of $M_W$ we show that we also have $B \in M_W$. And we forget about $\xi \lt (2^{|\gamma|^{\lt \kappa}})^{+L[A]}$.
$(2)$ We somehow conjure up some $A \in P(P(P_\kappa\gamma))$ such that either $L[A] = L[B]$ or at least $(2^{|\gamma|^{\lt \kappa}})^{+L[A]} = (2^{|\gamma|^{\lt \kappa}})^{+L[B]}$.
If any of the above arguments get completed, then it is easy to show that $\xi \lt j_W(\kappa)$ because we have $(2^{|\gamma|^{\lt \kappa}})^{+L[A]} \le (2^{|\gamma|^{\lt \kappa}})^{+M}$ and also $2^{|\gamma|^{\lt \kappa}} \lt j_W(\kappa)$, and we are done by the inaccessibility of $j_W(\kappa)$ in $M$.
To sum it all up, how can we fix the above argument? Or is there another way to produce such an $A$?

Edit I:
This exercise is from Kanamori's "The Higher Infinite", $2$nd edition, page $306$.
 A: Without loss of generality we can identify $\gamma$ with $\lambda=|\gamma|$.
Let $A\subseteq \mathcal{P}_\kappa \lambda$ be an code of a well-ordering of $\mathcal{P}_\kappa\lambda$: note that the map
$$\phi: (x,y)\mapsto \{\alpha\cdot 2\mid \alpha\in x\}\cup \{\beta\cdot 2 +1 \mid \beta\in y\}$$
is a canonical bijection between $\mathcal{P}_\kappa \lambda\times \mathcal{P}_\kappa \lambda$ and $\mathcal{P}_\kappa \lambda$, so we can encode a well-ordering of $\mathcal{P}_\kappa \lambda$ into a subset of it.
Let $B\subseteq \mathcal{P}(\mathcal{P}_\kappa \lambda)$ be a code of an order-type of $\xi<(2^{\lambda^{<\kappa}})^+$ given as follows: let $B_0\subseteq \mathcal{P}(\lambda^{<\kappa})\times \mathcal{P}(\lambda^{<\kappa})$ be a well-ordering of ordertype $\xi$. Fix any bijection $f:\mathcal{P}_\kappa \lambda\times 2\to \mathcal{P}_\kappa \lambda$. Then the map
$$(X,Y)\mapsto \{f(x,0)\mid x\in X\}\cup \{f(y,1)\mid y\in Y\}$$
defines a bijection between $\mathcal{P}(\lambda^{<\kappa})\times \mathcal{P}(\lambda^{<\kappa})$ and $\mathcal{P}(\lambda^{<\kappa})$.
Take $B_1=f^"[B_0]$. 
Moreover, there is a unique isomorphism $g:(\mathcal{P}_\kappa\lambda,A)\cong (\lambda^{<\kappa},\in)$. Finally let 
$$B = \{X\in \mathcal{P}(\mathcal{P}_\kappa\lambda) \mid g^"[X]\in B_1\}.$$
Let $W$ be an ultrafilter such that $B\in M_W$. Note that $A,f,g\in M_W$ and $\mathcal{P}_\kappa\lambda$ is absolute between $V$ and $M_W$. By reversing our construction of $B$, we have $M_W\models |\xi|\le|\mathcal{P}(\mathcal{P}_\kappa\lambda)|$. Therefore $M_W\models \xi < (2^{\lambda^{<\kappa}})^+$.
