I know of no evidence that individuals use both [ˈæntɪpɔʊd] and [ænˈtɪpədijz], though I also can't say that I've even ever heard anyone use "antipode". I don't control the meaning / use of "antipode", but the words are not in a regular singular / plural semantic relationship, as "cat / cats", or "mouse / mice" are. You expect a regular pronunciation relationship between regularly derived inflectional singular / plural pairs. However, there are also word pairs that stand in an etymological singular / plural relationship that don't have a synchronic singular / plural inflectional relationship, such as opus, opera. We treat these as separate words (and the Romans are responsible for the changes in pronunciation).
The two words are thus unrelated in Modern English. Antipode simply follows the regular rules for orthography-to-pronunciation. Perhaps the first person who coined the singular didn't know how to pronounce "antipodes", and followed the regular rule. The pronunciation of "antipodes" follows alternative rules of pronunciation that are relevant to words from classical languages. In the present case, Latin is a conduit for Greek ἀντίπους, underlyingly /anti-pod-s/ but even in Ancient Greek, the inflection of "foot" was irregular.
As you will observe in indices, theses, vortices, there is a special rule for pronouncing es in Latinate plurals. I expect that many people pronounce "antipodes" as [ˈæntɪpɔʊdz], if they are unaware of that rule (they probably don't teach it in school anymore), and the history of the word. (Similarly, hammamelidanthemum is mis-stressed with alternating stress starting with the first syllable in a certain famous stress dissertation, since the author was at the time unaware of the etymological makeup of the word).
It seems that the classical words with "es" plurals either derive from Greek is-stem nouns, e.g. thesis, analysis, synapsis, crisis, which had the plural -ɛις, or were Latin third declension nouns with the ending -ēs in the relevant words. Either source would yield the plural [ijS].