This depends slightly on the precise interpretation of your question, but I think your scenario can be generically described as a problem 'COMPUTE Y' where given some universally fixed polynomial time algorithm $T$ and polynomial $p$, on input $\langle x, 1^n \rangle$, output a string $y \in \{0,1\}^{p(n)}$, such that $T(x,y,1^n)$ outputs 1, and $y$ always exists for all possible $x$.
One question then might be whether a polynomial time algorithm for 'COMPUTE Y' implies $P = NP$
In this case, assume you can solve (say) 3SAT in polynomial time with a constant number of calls to an oracle that solves 'COMPUTE Y', i.e. some algorithm $A$ where $A(\phi) = 1$ iff $\phi$ is satisfiable, $A(\phi)=0$ otherwise. Flip the output bit to get $\bar{A}$, an algorithm where $\bar{A}(\phi) = 0$ iff $\phi$ is satisfiable and $\bar{A}(\phi) = 1$ if $\phi$ is unsatisfiable.
Convert this algorithm $\bar{A}$ (which uses an oracle for 'COMPUTE Y') into a nondeterministic algorithm (that uses no oracles) by simply replacing each oracle call with a nondeterministic guess of $y$ that you can check with a call to $T$. Now you have a nondeterministic algorithm which successfully decides unsatisfiable 3CNF instances, so $NP = coNP$
As an aside, if $NP = coNP$, that implies that all $NP$ complete problems (like $k$-clique or 3SAT) have slight variations whose decision problem is easy (always 'yes') yet whose search version is $NP$-hard