The R443 FD is designed for (rapid fire) mountain bike indexing/cable-pull. The cage is shaped to use with a road bike crankset (thus the "R" in the product code). For the best performance, you want to use the Tiagra flat-bar shifter with the matching front derailleur.

You could use a mountain bike 9-speed front shifter instead of the Tiagra, but that definitely would not work with your current road front derailleur. You would need to use a mtb front derailleur in that case. The problem with a mtb front derailleur is that the cage is the wrong shape for a 50t or 52t big chain ring, so then the crank wouldn't match up... thus the need for a special derailleur like the R443 which has mtb cable-pull but a cage shaped for a road crankset.
You could use a 9-speed mtb shifter with the R443, but the Tiagra version is better because it has additional trimming positions.

So, the best match is a shifter and front derailleur from the same Tiagra flat-bar group. However, if you just buy the shifters mentioned above and keep your current road-specific front derailleur, it can be made to work due to the additional trimming positions on SL-R440-9... it will just take a little more fine-tuning to get the shifting dialed in. Best bet is to just buy the matched set and be done with it, the FD is cheap enough on fleabay.
Alternatively, you could buy a SRAM twist-grip style shifter that is indexed for Shimano, as the left hand shifter isn't indexed and can be made to work with any front-derailleur.

You never said what size your big ring is... 443 FD is for 50t chainring while FD-R453 is for the 52t.