[His first instinct to answer "Archer", even after a while, but at least he's starting to get more comfortable and doesn't seem like it's a forced answer.]

Either is fine, but I'm going to say Emiya to get used to it properly.

[But if some people want to continue calling him Archer (or Narwhal, he supposed...) he wouldn't complain.]

Make sure to have something personal as a gift to appaese her. The more personal it is the more likely she'll answer with a good mood.