I have used my Kenko Pro 1.4 TC with my 400mm F5.6, but not often. I've had reasonable success down to 1/200sec@F10.0. If I'd noticed in camera that the shutter speed was only 1/200,

I probably would have altered the aperture to F8.0 to give more speed, but the shot wasn't too bad in the end.
This was using my monopod. I doubt I could have hand held at that speed successfully. If I remember right, I was also leaning on a fence post as well, so the lens was quite well supported.
Provided it's a good bright day there's no reason why you shouldn't get good results, the lens is certainly sharp enough to give good results with a 1.4 X TC fitted, provided you can manually focus well.