I own both of those lenses and considered disposing of the 300/4L IS and using my 1.4xTC II on my 70-200/2.8L IS II when I needed the extra reach. For the record, I shoot them on a 1D Mark IV but I suspect the relative performance would hold true on any body.
The 70-200/2.8L IS II is an outstanding lens but with the 1.4xTC behind it, the 300/4L IS outperforms it for sharpness, colour and contrast as well as AF speed. Remember that Canon states the TC will slow AF by approximately 50%. This was done by design to provide the most accurate AF.
Also, if you already have the 1.4xTC, keeping the 300 gives you a 420/5.6 lens as well. That can be a pretty good combination. Here is a shot I took hand held with the 300/4 + 1.4xTC a couple of weeks ago. It's about a 50% crop.Be certain to click on the image to bring it up full size.
I'm keeping both.
Hope this helps.