The following example relates to the second part of the question while echoing back the example $R = \mathbb{Z}[X, Y]$ and $q = (X^2 + Y^2 - 1)$ discussed above:
Let $k$ be a finite field and let $q$ be any ideal of $R = k[X, Y]$, then the natural map $SL_n(R) \rightarrow SL_n(R/q)$ is surjective for all $n \ge 2$.

This is Theorem 1.7.(2) of "On the groups $SL_2(\mathbb{Z}[x])$ and $SL_2(k[x,y])$" by F. Grunewald, J. Mennicke and L. Vaserstein, 1994 (MR1276133).

In the spirit of the requirements of the first part of the question, I would ask whether $2$ is the smallest Krull dimension we can get for a ring $R$ generated by finitely many elements and for which surjectivity of the reduction of matrix coefficients modulo $q$ fails for some ideal $q \subset R$.

The case of $\mathbb{Z}[X]$ is somehow settled by Theorem 1.7.(1) of the same paper: for $R = \mathbb{Z}[X]$, the image of $SL_n(R)$ in $SL_n(R/q)$ is of finite index for every $n \ge 2$. In some sense, it is optimal since F. Grunewald et al. have a recipe to build quotients of $\mathbb{Z}[X]$ with non-trivial $SK_1$ (see Proposition 1.9 of the same paper and this MO post) whereas $SK_1(\mathbb{Z}[X]) = 1$.

As for the general part of the question, the group $SK_0(q)$ (see Definition II.2.6 and Exercise III.2.1 of C. Weible's K-book) is the natural obstruction to the surjectivity of coefficients reduction modulo $q$. You may argue that's kind of tautological though.

**Addendum:**
T. Goodwillie's example originates from Example 13.5 of "Introduction to algebraic $K$-theory" by J. Milnor, 1971 (MR0349811).
To some extent, it is also discussed in "Serre's problem on projective modules" by T. Y. Lam, 2006 (MR2235330), see in particular Proposition I.8.12 and Remark I.8.14.