I'm not aware of any definitive information in this respect, unlike that which is available for other famous saxophonists. The photograph evidence shows that Carney appears to play Woodwind Co mouthpieces, but it also shows that in 1954 or thereabouts he had a mouthpiece with a metal band around the neck, which could have been a Conn mouthpiece or something similar.
I am not aware of any evidence which tells us that the model of Woodwind Co mouthpiece he used at any particular time was a Sparkleaire as opposed to the regular steel ebonite models.
I'm also not sure which mouthpiece he used in later years (when he played a Conn low-A model). The mouthpiece, whilst still quite fat, looks a little less fat than the typical Woodwind Co models.
Also, in a 1956 Conn full-page advert (which I have framed on the wall right here), which features Carney, there is a mouthpiece which appears to be a little different to the regular Woodwind Co mouthpieces (of which I have many). It may be an artefact of the light and the photo, but the profile of the mouthpiece has something: where the main body of the mouthpiece reduces to the diameter of the neck (which is slightly tapered), there appears to be a lip.
You can see a poor resolution image of the same advert here:
As you can see (better on the real paper image), after the ligature the slight taper of the body of the mouthpiece suddenly increases to create a bit of a lip before it reduces quickly down to the start of the neck. I have never seen any mouthpiece like that, and I want to know what it is!