Wrestling firemans carry
It sounds like you are talking about some variation of the wrestling fireman's carry. I am not sure whether actually picking the opponent up in wrestling changes the classification. The instructor in this video definitely lifts completely in one example, but in another appears to proceed directly to the sacrifice.
Judo uki waza or yoko otoshi (formal), kata guruma (informal)
In judo, this has been called kata guruma, though if you are using a sacrifice technique, this is not the current formal classification. For understanding arcane formal judo classification, the best source is Kodokan Judo: Throwing Techniques by Toshiro Daigo, Kodansha, Tokyo, Japan 2005. Toshiro Daigo, the current head of the Kodokan, explains a sacrifice in the kata guruma position would be either uki waza or yoko otoshi.
Daigo p. 232 discusses "uki-waza from the form of kata guruma":
After a study by the Kodokan Waza Research Institute (April 1, 1997), it was decided that this technique is related to uki-waza and not kata-guruma, because the concept of the throw is based on the principle of yoko-sutemi-waza [side sacrifice technique].
Daigo p. 235:
It is extremely difficult to clarify the differences between these techniques [yoko otoshi and uki waza], and is a matter for the judges.
The points of judo throw classification can be quite complicated and opaque. I would recommend against investing too deeply in understanding the Kodokan current thinking about the differences between uki waza and yoko otoshi. In everyday usage, I would also expect to see this called kata guruma as it is in the linked youtube video, as the official word on this is only from 1997.