when Klein was in Tingen he always chose to pair coffee with three sugars and milk, or a black tea with lemon. Audrey notes that when he's acting as Dwayne he has coffee with three sugars and without milk, or tea and lemon.
when offering drinks to Klein during a therapy session, he chooses tea with lemon.
this is well after he was told to wear a thinner mask.
my theory is:
he started drinking coffee without milk for the Dwayne persona, thinking it was more mature, but refused to compromise on the sugar.
he knew Audrey would probably not add milk to his coffee as she would have noted his 'preference' by now.
he chose to be true to his heart and therefore picked tea, which he had always had with lemon.
true to this, despite only asking for black tea, Audrey still requested a tea with lemon from her maid