Hardness of approximating fractional chromatic number on bounded degree graphs
Yes.

If I understood correctly, the proof of Theorem 1.6 in Khot (2001) establishes that it is NP-hard to distinguish between the following two cases, even if we focus on bounded-degree graphs of sufficiently high degree:


There is a $k$-colouring.
The ratio of the number of vertices to the maximum size of an independent set is at least $k^{\log(k)/25}$.


From the perspective of fractional chromatic number these two cases are:


The fractional chromatic number is at most $k$.
The fractional chromatic number is at least $k^{\log(k)/25}$.


Now we must remember that we need sufficiently high degrees (as a function of $k$). But as far as I can see, the proof has, e.g., the following convenient corollary that might already be sufficient for your purposes:


Given any constant $\alpha$, there are constants $\Delta$ and $c$ such that the following problem in NP-hard: given a graph $G$ of maximum degree $\Delta$, decide whether the fractional chromatic number of $G$ is at most $c$ or at least $\alpha c$.


An of course this already implies that there is no PTAS, unless P = NP.