I'm looking for problems that are hard to solve in FPT time but has an approximation algorithm. That is, problems that are:
R2. Admit a (preferably constant) approximation algorithm in FPT time.
The problem I'm familiar with is counting the number of simple paths of length $k$ in a graph. It is known to be #W-hard, but admits a $(1+\epsilon)$-approximation in FPT time (for any constant $\epsilon$).
Also interesting would be problems that satisfy R1 and R2, and also:
R3. There exists $\epsilon>0$ such that the problem is not $(1+\epsilon)$ approximable in FPT time (unless W=FPT).
What other problems satisfy R1 and R2, and possibly R3?