I'm sure I wouldn't be telling you something that you don't know if I say that the addition of a converter optically alters the lens- it is, in effect, a new lens, possibly (almost certainly) with different distortions and vignetting than the lens alone. And every new model TC would be different than its predecessor. So, for complete coverage Adobe would have to make separate profiles for every possible combination; but I see that for the 30+ Canon EF lenses over 70 mm. in the LR menu, only 8 have additional profiles with a TC - including the 200-400 f4 with its dedicated TC. The other 7 (these lenses:
70-200 f2.8L IS II
100-400 f4.5-5.6L IS II
300 f2.8L IS II
400 f2.8L IS II
400 f4 DO IS II
500 f4L IS II) have two additional profiles, for 1.4x and 2x, but the TCs are only identified (in the body of the profile as seen in WordPad) as either 1.4x or 2x. IOW, there are are not separate profiles for the 1.4, 1.4 II and 1.4 III. So there are two questions: Were all the profiles made with the original Canon TCs? And how does your 1.4 III identify itself to the camera and is it described in the EXIF as 1.4x III? If its EXIF tag is different from that of the old 1.4, that could be why LR is not automatically finding a match; but that is a good thing because it prevents the use of an inaccurate profile.
You could make your own profile with the software provided by Adobe (I remember kirkt having a go at it several years ago, you could PM him for advice) or, alternatively, Adobe has a site where community members post homemade profiles and also provides an app for searching and downloading from there (I once got a profile for an antique Sigma 15-30 there). Or you could use the LR-provided profile and eye-ball any changes it needs with the manual Distortion and Anti-Vignetting sliders.