Approximation ratio when optimal solution is $0$
That's right.  Such a problem is not approximable to within any constant factor large than 1 (unless P=NP).

Is there a way around this?  Yes, there are several possibilities:


Use a different measure of approximation, rather than multiplicative factor.  For instance, you could consider its additive approximability.
Re-define the problem so it cannot have output of 0.  For instance, let $Pc(G)=Pb(G)+1$, i.e., $Pc$ is the problem of computing the minimum number of edges to remove from a graph so it becomes 3-colorable, plus one.  Then there is no such barrier to achieving some constant approximation factor. (There might still be other barriers, of course.)  Suppose you found that $Pc$ has a 2-approximation.  Then you could unfold what that implies for $Pb$.