However the 70-300 is no where near as sharp as the 70-200.
In what parallel universe? Unless you're talking about the 70-300 non-L, in which case, certainly. But, the OP has the 70-200/2.8 IS MkI, and the 70-300mm L is a fair match for sharpness across the range when the 70-200 is stopped down to comparable apertures (and with both wide open, the 70-300L is sharper).
He currently has a 70-200 F2.8 IS which is a magnificent lens. It works well with both the 1.4x and 2.0x tc and will autofocus with any camera with either converter giving a usable range of 70-400mm
Allow me to edit for content: He currently has a 70-200 F2.8 IS which is a
magnificent very good lens. It works
well sort of ok with
both the 1.4x and
delivers soft images with the 2.0x tc and will autofocus with any camera with either converter giving a usable range of 70-
400mm280mm, or 70-400mm if you have a high tolerance for mushy imagesIf the OP had the 70-200/2.8 IS
MkII, it would be a different story. But the MkI is the least sharp of the three 70-200/2.8 zooms (and the f/4 IS is sharper, as well), and the MkI does not handle teleconverters well at all.
For example - you can see that
comparing 70-200/2.8 IS at 280mm f/5.6 (stopped down) to the 70-300 L at 300mm f/5.6, the latter is sharper. As for the 2x TC, here's
what that does to IQ. Hmmm...looking at that, the effect in the center is not as bad as I anticipated - but that's because the bare lens at 200mm f/2.8 is not that good in the center (at least, when
compared to the MkII).
So, compared to the 70-200/2.8 IS, the 70-300L would be an IQ upgrade if the OP needs the 200-300mm range, but the tradeoff is a loss of one stop of light.