A word for someone who feigns ignorance

Solution 1:

I think dissimulate would work:

verb (used with object), dis·sim·u·lat·ed, dis·sim·u·lat·ing.

to disguise or conceal under a false appearance; dissemble: to dissimulate one's true feelings about a rival.

verb (used without object), dis·sim·u·lat·ed, dis·sim·u·lat·ing.

to conceal one's true motives, thoughts, etc., by some pretense; speak or act hypocritically.

In your example A is dissimulating. or alternatively you could use the synonym in the definition, A is dissembling.

But if you are looking for an adjective rather than a verb, you could say the person is disingenous -- in the sense of falsely or hypocritically ingenuous.

adj lacking in frankness, candor, or sincerity; falsely or hypocritically ingenuous; insincere: Her excuse was rather disingenuous.

Solution 2:

I think you are describing passive-aggressive behavior.

Passive–aggressive behavior is characterized by indirect resistance to the demands of others and an avoidance of direct confrontation.

Here's a description of one of the symptoms of passive-aggressive behavior:

On the other extreme, insincere compliments are a very sneaky way to throw in some nasty insults. They are hoping that you will react badly to the insult. The intention being that if you say anything about the cheap insult, they will tell you that you misinterpreted them, and they were only trying to pay you a compliment. Then you doubt yourself and, if anyone else was present, your behaviour looks unreasonable.