Assuming a new identity

Well, that depends on how willing MC is to go along with it. If he agrees with the need to change names, then he would use the new one, wouldn't he?

In the narration it would make sense for the main character to continue to refer to the other character by her original name, but in dialogue it probably would make more sense to use her new name. If a name/identity change was essential, then using the old name would be foolhardy in many situations.

Basically, in narration and in dialogue away from the problem area [school], the character retains her old name. The situation necessitated the name/identity change, so I'd just suggest you take the same general approach with the rest of the story and only use the new name/identity when necessary.

Edit: But, it also depends on what sort of person the main character is, as Judg said. If the main character is of a brash sort, throwing caution to the wind, then name/identity slip ups or intentional stupidity, using the old name/identity at the wrong moment or in the wrong company would make sense. It hinges on what the main character's personality is and the slip ups could make for interesting turns of events.

IMO the MC would use her new name when conversing with her in public places where others might overhear and use her real name when they were alone. Unless the MC doesn't really want to protect her identity.
Linnea

Part of it depends on what the MC thinks of the whole name change. She may get to the point where she accepts this new name completely.

An example: my friends and family stopped referring to me by my maiden name after I got married. As far as they are concerned, I'm now Clair Dickson. They fully accept that I changed my name and only refer to me that way. (Sometimes people that know me forget that I am still part of the same family as my brothers-- it's amusing.)

The MC might even begin to think of her friend by the new name, but may still recall the days when Lea still was Violet. If so, she'd make the switch to referring to her entirely by the new name (consistently.)