I gather the front grip shift is just a friction shifter so I would expect it should work. Either wind the limit screw in so the derailleur is in the middle position or use a screwdriver to jam the front derailleur somewhere towards the middle position and install the cable. Re-adjust or remove the screwdriver - now the cable should be slightly tight in the small ring, see if you have enough travel.
Otherwise you might need a MTB front derailleur, they have different travel than road derailleurs. But I expect it should work the way it is.