Design FIR with linear phase response
What you want is basically a delay
$$x(t-\tau) \Rightarrow e^{-j\omega \tau}$$
If you can live with $k$ being quantized you can simply use a single tap delay FIR filter. If you need more granularity you need to implement a fractional dealy. I would cascade a shot fractional FIR filter with a long single tab bulk delay filter.
If $k$ is positive the filter becomes a non-causal. The best way to deal with this depends a bit on your application. If you do off-line processing, you just need to delay all other signals paths by the same amount.