Effect of exogenous parameter on three player SPNE profits
Let's solve by backward induction.

In stage 3, player $A_3$ solves:
$$
\max_{p_1, p_2} u_3(p_1, p_2, q_1, q_2, x).
$$
Given the assumptions, $A_3$ will set: $p_1 = \overline{p}$ and in general $p_2$ will be a function of the $q_1, q_2$ and $x$, denoted by $p_2(q_1, q_2, x)$. For a solution to exist, you have to assume that $u_3$ is not unbounded in $p_2$.

As $u_2$ is independent of $p_1, q_1, x$, In stage 2, Player $A_2$ solves:
$$
\max_{q_2} u_2(q_2, p_2(q_1, q_2, x)).
$$
$u_2$ is increasing in $q_2$ and $p_2$. If $p_2(q_1, q_2, x)$ is increasing in $q_2$ then $A_2$ will set $q_2$ as big as possible, so there is no optimal solution. Else, there might be some optimal level of $q_2$, say $q_2(q_1, x)$.

In stage 1, player $A_1$ solves:
$$
\max_{q_1} u_1(q_1, \overline{p}).
$$
As $u_1$ is increasing in $q_1$ there will probably be no solution to this. So likely this problem is unbounded. If we assume that $q_1 \le \overline{q}_1$ for some fixed value $\overline{q}_1$, then $A_1$ will set $q_1 = \overline{q}_1$.


Going back to the utility of player 3, we therefore see, after substituting:
$$
u_3 = u_3(\overline{p}, p_2(\overline{q}_1, q_2(\overline{q}_1, x),x), \overline{q}_1, q_2(\overline{q_1}, x), x).
$$
It seems like the change in $u_3$ due to a change in $x$ is ambiguous. As it also influences the optimal choice of $q_2$ (of player $A_2$), and $p_2$ of player $A_3$.