Let me first explain what I mean by the "classical limit". For quantum group invariants of links and webs (such as colored Jones polynomials), the "classical limit" means the limit $k\rightarrow +\infty$, or equivalently $q \rightarrow 1$. In this limit, quantum invariants become insensitive to the embedding of links and webs inside $\mathbb{R}^3$ (or $S^3$), and they count the number of certain edge colorings. So, in this sense, "quantization" (deforming $q$ away from $1$) is what makes those invariants sensitive to embedding.
Categorification deforms another variable $t$ away from $-1$ (if we just focus on the Poincare polynomial). Hence, starting from the "classical invariant" counting certain edge colorings, we have two different deformations (see the diagram below), and it is natural to wonder if we can complete the square by filling in the "???" part. $$\begin{matrix}??? & \xleftarrow{?} & \text{categorified quantum invariant} \\ \quad\quad\downarrow\small{t=-1} & & \quad\downarrow\small{t=-1}\\ \text{classical invariant} &\xleftarrow[]{q=1} &\text{quantum invariant}\end{matrix}$$
That is, I am curious if there is a certain specialization "?" (which I want to call the "classical limit") of the categorified quantum invariant (e.g. Khovanov homology) such that it is independent of the embedding of links and link cobordisms inside $\mathbb{R}^3$ and $\mathbb{R}^3\times \mathbb{R}$, and when specialized to $t=-1$ it becomes the classical invariant. Ideally, "???" must be a certain type of 2d TQFT.
Certainly, the naive guess of $q=1$ doesn't work in this case, as even in such specialization, Khovanov homology is still sensitive to embedding. Therefore, my question can be summarized as : Is there a specialization of Khovanov homology (or any other categorified quantum invariants) of type $f(q,t)=0$ for some polynomial $f$ such that the invariants become insensitive to the embedding under such specialization?