The name "Tsar Bomba" is actually a nickname originating from the West. The bomb was officially known as RDS-220, and was called Big Ivan by its designers. You are right though, this name would never have been used by the soviets, yet has stuck since the collapse of the soviet union.
Tsar-bomba is extrapolated from Tsar-kolokol and Tsar-pushka in the same way as Tsar-tank. Here Tsar stands for "King size" in a mildly ironic manner.
I don't actually know when the "Tsar Bomba" name got attached to it, but it wasn't used in its time at all. The US press labeled it as "the 100 MT bomb" usually when they spoke of it. The Soviet and American code-names for its test were not released until after the war ended. One newspaper tried to get the term "K-Bomb" adopted (for Khrushchev) but nobody went with it. I have been trying to figure out who started calling it the Tsar Bomb (in particular, whether it was American or Russians who started it). Google Ngrams has zero entries for it before 2000 (which is not entirely reliable but helps emphasize how recent this term is), and none at all in their Russian corpus.
As for whether it was antithetical or not — it would not have been a polite name for it at the time. The Tsar Bomba name is, as other have noted, an allusion to other large-but-impractical weapons developed primarily by the Russian Imperial state. The 100 Mt bomb was large but the Soviets would have denied it was too impractical to use. The CIA would have secretly agreed — they worried that the Soviets would find (relatively simple) ways to slim it down a bit and put it on a super-heavy ICBM, and they were awed by the extreme distances its thermal radiation would travel when it was detonated high above the ground. The public US reaction was to say it was too impractical, just a stunt, etc., but the private one was more concerned.
Source: I have been researching the US government's public and private reactions to the Tsar Bomb detonation for a little while now, and am eventually going to write an article on it.