thinking about invariants of 2-knots ... is there an obvious map from $H_2(X)$ (or from part of it) into $\pi_1(X)$?
Where in the derived series of $\pi_1(X)$ would the image of $H_2(X)$ live?

Have you heard of things like Dwyer's filtration on $H_2$ and Stallings/Cochran-Harvey theorems about the lower central/derived series? I cannot resist quoting the opening sentence of Krushkal's paper:

"The lower central series of the fundamental group of a space $X$ is closely related to the Dwyer’s [D] ﬁltration $\phi_k(X)$ of the second homology $H_2(X;\Bbb Z)$."

The Dwyer subspace $\phi_k(X)\subset H_2(X;\Bbb Z)$ is deﬁned as the kernel of the composition $$H_2(X) \to H_2(\pi_1X) \to H_2(\pi_1X/\gamma_{k-1}\pi_1X).$$ The gamma notation for the lower central series runs $\gamma_1G=G$, and $\gamma_{k+1}G=[G,\gamma_k G]$.

Freedman and Teichner showed that

$\phi_k(X)$ coincides with the set of
homology classes represented by maps of closed $k$-gropes into $X$.

Now their $k$-gropes are gropes of *class* $k$, but you can also do *symmetric* gropes
of *depth* $k$, or whatever they are called, to get into the *derived* series. Which series to choose, and how to make use of it? The lower central series has a long standing record of applications in knot/link theory due in part to Stallings' theorem (1963, also rediscovered by Casson; note Cochran's topological proof):

Let $\phi:A\to B$ be a homomorphism that induces an isomorphism on $H_1(−;\Bbb Z)$ and an epimorphism on $H_2(−;\Bbb Z)$. Then, for each $n$, $\phi$ induces an isomorphism
$A/\gamma_nA\simeq B/\gamma_nB$.

Dwyer (1975) extended Stallings’ theorem by weakening the hypothesis on $H_2$:

Let $\phi: A\to B$ be a homomorphism that induces an isomorphism on $H_1(−;\Bbb Z)$. Then for any $n\ge 2$ the following are equivalent:

• $\phi$ induces an isomorphism $A/\gamma_nA\simeq B/\gamma_nB$;

• $\phi$ induces an epimorphism $H_2(A;\Bbb Z)/\phi_n(A)\simeq H_2(B;\Bbb Z)/\phi_n(B)$.

Now if you do want the derived series rather than the lower central series, take a look at the work of Cochran and Harvey who had a series of papers about the analogues of the Stallings and Dwyer theorems for torsion-free derived series. In fact their motivation was also knot theory.

Also Mikhailov has some other generalization of the Dwyer filtration (see also his book with Passi in Springer Lecture Notes in Math) though I'm not sure if this has had any application to knots.

In fact, I think Mikhailov, and possibly Orr and Cochran, also did something about the transfinite Dwyer filtration, which might be not unrelevant to knots and links (at least this is where the transfinite business originated from, in papers by Orr and Levine in the 80s).