Is there any way, using some form of CCA or whatever the sort, to recover the primes/private exponent of the RSA key?
TLDR: No, save for a breakthrough or a leak.
Whether $N$ can be factored (in time polynomial to $\ln N$ with non-vanishing probability) given $(N,e)$ and an RSA decryption blackbox (assumed free of side-channel leakage) is one of the most thought at open problem in cryptography, and to a lesser degree in number theory.
Proving either yes or no would make the news, even if restricted to an heuristic algorithm, plausible mathematical hypothesis, a polynomial of degree so high that the method is impractical, and a particular choice of $e$. I'm thinking of $e=3$, or $e=N$ (the Clifford Cock's cryptosystem, which predates RSA), or on the contrary random odd $e\in[0,N])$.
A proven fact is that extracting any working private exponent $d$ of size upper-bounded by a polynomial in $\ln N$ allows to extract a factor of $N$, at least heuristically and in heuristic polynomial time, and in practice allows to factor $N$.
Another experimental fact is that if we probe a physical implementation of such a black box, that has a public design, long enough, with sensitive-enough sensors, and pour enough brainpower and a little CPU power on the thing, we manage to extract $N$. This is one of several understandable reasons why such black boxes are not made of public design.
BTW: experimental cryptanalysis with quantum sensors is the one unobstructed way for experimental cryptanalysis to go quantum.