Such homotopies are given by the $\smile_i$-products. Steenrod gives explicit formulas, IIRC, in [Steenrod, N. E. Products of cocycles and extensions of mappings. Ann. of Math. (2) 48, (1947). 290--320. MR0022071], but the easiest is to prove they exist using acyclic models.
(Maybe Steenrod only deals with $\mathbb Z_2$ coefficients? I don't have access to the paper now :( )
Since no one else answered, I'll take a stab at it, although I am not sure I'll be answering the question you are asking, since some of your notation confuses me: but here goes.
There are lots of ingredients: First, the DeRham map, defined by assigning to a differential $k$-form the cochain which integrates the form over a smooth $k$-simplex
for any smooth manifold $M$ yields a ring isomorphism $H^*_{deRham}(M)\to H^*_{sing}(M;R)$. The ring structure is wedge product on forms and cup product on singular cohomology. You can find proofs in Warner or Wells. Stokes' theorem is the fact that the DeRham map is a chain map from differntial forms to the singular complex.
The fact that the DeRham map is a ring map is a bit tricky, but morally follows because "what else could it be?", i.e. naturality says any natural pairing on the singular chain complex that might induce the cup product must induce the cup product on cohomology. Precise statements can be found in algebraic topology texts.
Next, given a submanifold $C^k\subset M^n$, or more generally a subvariety with singularities of real codimension at least 2, A triangulation of $C$ determines a singular $k$ cycle in $M$ denoted by $[C]\in H_k(M)$.
(inverse) Poincare duality defines an isomorphism $P:H_k(M)\cong H^{n-k}(M)$. "Intersection theory" says that
if $C_1, C_2$ are submanifolds of $M$ of complementary dimensions ($k_1+k_2=n$), then the transverse algebraic intersection number of $C_1$ and $C_2$, $[C_1]\cdot [C_2]\in Z$ equals $(P([C_1])\cup P([C_2]))\cap [M]$, where $[M]\in H_n(M)$ denotes the fundamental class. A proof can be found in most algebraic topology books, eg Greenberg-Harper. It relies ultimately on some form of the Thom isomorphism theorem, whose standard proof is a spectral sequence proof.
In your case $C_1$ and $C_2$ are complex subvarieties, and hence every transverse intersection is positive, thus the algebraic count (with signs) equals the geometric intersection number, and is non-negative.
Next, "$\cap [M]$" corresponds to integrating over $M$ via the DeRham isomorphism. This follows from the identification of the cap product with the Kroneker pairing on top cohomology, and the definition of the DeRham map (e.g after triangulating $M$). Thus if $\eta_1, \eta_2$ are closed forms whose cohomology classes are sent to $P[C_1], P[C_2]$ by the DeRham isomorphism, then using the fact that $\wedge$ maps to $\cup$, $$\int_M \eta_1\wedge \eta_2=(P[C_1]\cup P[C_2])\cap [M]=[C_1]\cdot [C_2]$$
Now, assuming this is related to what you wanted to know, the $2\pi i$ you mention are factors that come into play in Chern-Weil theory, which is the set of results that express characteristic classes of bundles over manifolds in terms of differential forms constructed out of the curvature form of any connection on the bundle. You dont need singular forms to prove these formulae: see for example Appendix C of Milnor-Stasheff for an elementary exposition.
I dont have GH here, but presumably they express $c_1(L)$ in terms of a divisor, i.e. the zero set of a section of $L$, and perhaps they want to insist it be holomorphic, and so the divisor might be a singular subvariety, with dual form a singular form.
Best Answer
Bott and Tu do this completely, in the de Rham theoretic setting of course.
Here's an alternate proof I have used when I teach this material, which I find slightly more clean and direct than using Thom classes in de Rham theory (which require choice of tubular neighborhood theorem, etc) and works over the integers.
Definition: Given a collection $S = \{W_i\}$ of submanifolds of a manifold $X$, define the smooth chain complex transverse to $S$, denoted ${C^S}_*(X)$, by using the subgroups of the singular chain groups in which the basis chains $\Delta^n \to X$ are smooth and transverse to all of the $W_i$.
Lemma: The inclusion ${C^S}_*(X) \to C_*(X)$ is a quasi-isomorophism, for any such collection $S$.
Now if $W \in S$ then "count of intersection with $W$" gives a perfectly well-defined element $\tau_W$ of ${\rm Hom}(C^S_*(X), A)$ and thus by this quasi-isomorphism a well-defined cocycle if the $W$ is proper and has no boundary. It is immediate that this cocycle evaluates on cycles which are represented by closed submanifolds through intersection count.
There are two approaches to show that cup product agrees with intersection on cohomology. Briefly, one is to take $W, V$ over $M$ and consider the special case of $W \times M$ and $M \times V$ over $M \times M$. There some work with the K"unneth theorem leads to direct analysis in this case. But this case is "universal" - cup products in $M$ are pulled back from ``external'' cup products over $M \times M$. A second proof given in https://arxiv.org/abs/2106.05986 uses a variant of the theory, where one fixes a triangulation or cubulation, and assumes $W, V$ transverse to those. There we explicitly see that these products do not agree at the cochain level (they can't since intersection is commutative, but non-commutativity of cup product is reflected in Steenrod operations), but Friedman, Medina and I show a vector field flow leads to a cobounding of the difference.