Parametrized reduction from 3-SAT to Independent Set to lower bound running time under ETH assumption
First, let me remark that $n=O(m)$ as in each clause only $3$ variables may appear, and that $m=O(n^3)$ without loss of generality as there can be at most $(2n)^3$ clauses up to repetition.

Your bound is not correct. Consider the case where some variable $X$ occurs positive in half the clauses and negative in the other half. This leads to roughly $\frac{m^2}{4}$ edges for $X$ alone. What is correct, however, is your suspicion that the parameter dependence is superlinear.

The reduction can be modified such that $|V|=3m+2n$ and $|E|=6m+n$. This would lead to a linear dependence of $k_{IS}$ on $k_{3-SAT}$. Consequently, with these parameterizations, independent set does not have a subexponential algorithm unless 3-SAT has.

However, there appears to be a further problem: The ETH uses a different parameterization. Namely just the number of variables, disregarding the number of clauses. I am not an expert in ETH. Possibly the sparsification lemma can help here.