Any of Canon's 1.4x extenders will work with any version of the 70-200mm. The 1.4x III offers improved AF tracking with the IS II lens and slightly lower CA. If you go for the IS II, the 1.4x III would be the best choice of Canon extender. If you go for a used IS Mk I, the benefit of the 1.4x III over the 1.4x II is less clear but still the better choice, all other things being equal.
Users of the Kenko Pro 300 DGX usually report very good results.
With any teleconverter, sample variation between it, the lens and the body may be a bigger factor so a body with AFMA would be an advantage.