In your position I would do both.
Those on this thread who advocate going FI only have obviously never worked as a Flight Dispatcher and experienced all the airline contacts one can make this way. I can testify that when it comes to getting the elusive interviews, it's simply priceless. I applied to a number of outfits for ages with no results (not even an acknowledgement) - but when I started passing CVs to crews on turnarounds, hey
pronto! I got invited to interviews within weeks. As in many other businesses, it's not (only)
what you know, but
who you know. Incidentally, you will also learn lots of useful stuff in ops/ground handling. Most large handling companies (Servisair, Aviance, Swissport, Menzies, etc.) will be happy to have you as a Flight Dispatcher, even part-time. That's for the airline exposure.
Then, by all means, do work as an part-time FI to build up your hours and keep current. Likewise, a number of flight schools/aeroclubs will be only too happy to have even a part-time FI.
That way, with careful planning, you can get the best of both worlds. A couple of my friends did exactly that, and they didn't have to wait long for the call, and the interviewers were impressed by the "can do" attitude.
Cheers
