To what character (or characters) does the title refer? Is the title used straightforwardly or ironically?

The title "Defender of the Faith" is an ironic one.
Traditionally, the title of Fidei Defensor (Defender of the Faith) was conferred to the King of England. Literally, it identifies a religious champion.
In Roth's story, Sergeant Marx finds himself in that role, albeit unintentionally. In fact, he ends up favoring Grossbart and his two friends unintentionally, as his true motivation is to give fair and equal treatment to all his men without favoring Jews. However, Grossbart manages to trap him into being a Defender of the Faith each and every time thanks to his persuasive abilities.
Grossbart, on the other hand, is a false Jew. Jewishness has no particular religious meaning for him, and he only uses it as an excuse to get the undeserved privileges he wants. He is posing as a Defender of the Faith, but the word faith actually has no meaning to him whatsoever.
In the end, Roth evolves into a true Defender of the Faith, even though he seems to be turning against it by using Grossbart's same tactics. In fact, he manages to restore the balance by getting Grossbart sent off to the Pacific instead of New York, which was the original plan before Grossbart used his exploitative skills in order to be sent to New York. Essentially Roth is telling us that, for once, if the end is worthy the means can be overlooked.
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