The Principle of Optimality and the Bellman Equation

Recall that the Principle of Optimality states that the solution to Our Bellman Functional Equation is the same as the solution to the sequential problem if:

Assumption 1: $Gamma(x)$ (our set of feasible values) is non-empty for all $xin X$.

Assumption 2:$lim_{trightarrowinfty}sum_{t=0}^infty beta^t F(x_t,x_{t+1}) $ exists for all $tilde{x}in Pi(x_0)$ ($Pi(x_0)$ being the correspondence of $x$ considering an initial $x_0$)

Assumption 3:$|V^*(x)<infty|$ for all $x in X$.

Assumption 4: For any $x_0$, there exists a plan $tilde{x}in Pi(x_0)$ such that $u(tilde{x})=V^*(x_0)$.

In your story here since $beta<0$ we know that this is bounded all we are doing is adding an additional constraint. This is equivalent to writing the legrangian: $$mathcal{L}=sum_{t=0}^infty{beta^tu(c_t)+sum_ilambda_{t,i}(F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}}$$

Again I may not be iterating the whole story because there are two sectors here but this is how I'd go about thinking about if the principle of optimality is satisfied.

Hope this helps

Edit: After looking at your addendum im guessing the proceedure would be the same

$$V_t(k_t)=max_{c_t}{u(c_t)+lambda [F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}]}+max_{c_{t+1},...,c_{T}}{sum_{t+1} [beta^{t+1}u(c_{t+1})+sum_ilambda_{it}[F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}]]} $$

or $$V_t(k_t)=max_{c_t}{u(c_t)+lambda [F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}]+V(k_{t+1})}$$