Optimal price function: application of calculus of variations
Not really an answer, but too long for comment. 

The $P$ in your 
$$y(u,v)^*= \frac{v-P(y^*)}{P'(y^*)} +u$$ 
expression from the insider's problem and the $P$ in the expression 
$$
\min_{P(\cdot)} \cdots
$$ 
from the market maker's problem should not be the same. That is not the definition of rational expectation equilibrium in this context.

The insider takes the pricing rule as given, per definition of rational expectation equilibrium. This means insider has optimal demand 
$$
y(u,v, P_1(\cdot))^*= \frac{v-P_1(y^*)}{P_1'(y^*)} +u
$$ for any pricing rule $P_1(\cdot)$. Then market maker minimizes 
$$
E[ ( y(u,v, P_1(\cdot))^* - u) \cdot ( v - P_2(y(u,v, P_1(\cdot))^*) )  ]
$$ over $P_2$---taking $P_1$ as fixed. It's a simultaneous game.
If the minimizing $P_2$ coincides with $P_1$, then $P= P_1 = P_2$ is an equilibrium pricing rule. Doing this in the one-period Kyle setting recovers Kyle's lambda, whereas your calculation would not.

The optimization problem you're solving is therefore not the correct one.

The insider optimizes with respect to a conjectured pricing rule $P_1$. In equilibrium, the market maker confirms the insider's conjecture, making the latter's expectation rational.

Your calculation assumes that the market maker acts after the insider. This is not the case in Kyle and I don't believe it's the case in Rochet and Vila either.