Prove that there exist infinitely many integers $x$ such that integer $P(x)=x^3-2$ is a sum of two squares of integers.

Ideally, I am looking for a proof method that also applies for other $P(x)$, such as, for example, $P(x)=x^3+x+1$.

For $x=4t+3$, $P=(4t+3)^3-2$ is $1$ modulo $4$. By a well-believed (but difficult) Bunyakovsky conjecture, $P$ is a prime infinitely often, and every prime that is $1$ modulo $4$ is a sum of two squares.

To find an unconditional proof, it suffices to find polynomials $A(t)$, $B(t)$ and $C(t)$ with rational coefficients such that $A(t)^2 + B(t)^2 = C(t)^3-2$. Are there any good heuristics to find such polynomials? Is there any computer algebra system that helps to guess a solution of a polynomial equation over $Q[t]$?

One way to guess $C(t)$ is to form a set $S$ of all integers up to (say) $10^5$ that are sums of two squares, and look for polynomials $C(t)$ such that $C(t)^3-2$ belong to $S$ for all small $t$. I then checked all polynomials of degree up to $4$ and coefficients up to $12$, and found, for example, a polynomial $D(t)=3 + 8 t + 12 t^2 + 8 t^3 + 4 t^4$ such that $D(t)^3-1$ is always a sum of two squares. But no $C(t)$ found in this range, and increasing the degree and/or coefficients makes the enumeration infeasible. Are there methods to find a polynomial with all values in the given set, that are better than enumeration?