First, that conversation we had the other day about "What if Ichi-chan met another idol who looked up to her"
Second, she's partially based on the "Lisa" design you showed me. When you described her personality, it got me thinking about how she was the polar opposite of Grisaria.
So, if Ichi-chan is 1 and Grisaria is 2, what would a "-2 girl" be like? Timid, modest, pale skinned, struggles to stand out. The idea kinda of evolved from there.
Despite being counterparts of the same character, this girl and Grisaria couldn't be more different.
Good question. I didn't actually give her a name since she's not really my creation.
In the story she is her own character. If the original "Lisa" artist is still around, I think he should be the one to decide if she should have a name. In any case, I'll leave that up to you guys, the experts.

But yeah, in the comic, she insists on wanting to help Ichi-chan get together with Oguri Cap. It's worth noting that the story takes place before Ichi-chan and Oguri Cap become a couple but the general idea stands. The purple eyed girl wants to be like senpai and is confident that she can help her with her love problems.
She says that it's because she "Wants to be on senpai's level" but her exact feelings towards Ichi-chan are a mystery. Maybe it's admiration, maybe it's a sick obsession, maybe it's love. Maybe she doesn't even realize she feels that way.
Finally, in the comic we see Ichi-chan give her a chance (she's a good girl like that) but she is yet to befriend the others. Maybe they'll get along, maybe not. They are so different from each other so who knows.
I intentionally wrote her to be "a wrench in the machine" who breaks the dynamic and creates new opportunities for writing. All four girls are sociable and confident, how about one who isn't? What kind of dynamic would come from that?
And of course, she's on her own journey of personal growth. She wants to be less timid and more confident in herself.
Maybe she has a name, maybe not. Maybe she'll become part of the group, maybe not. Maybe she'll achieve her dreams, maybe not...
I've got some ideas for omake but I'll save them for later. Personally I took a liking for her because there are a lot of comedic scenarios that come to mind (specially regarding Belicose).
I'm also curious about how the original artist will feel about it.

I specifically released them once a day and used "Lisa" having white eyes when she's in her "number one fan" mode to hide the purple eyes.
So you can only tell that she's based on "Lisa" at the very end, when she calms down.
One more girl with yellow eyes and we have a full set. Twinkle Senshi
the filename LCA just means "Lisa Concept Art" btw.
There isn't a rule about it. Separated characters, counterparts. It's up to the writer. I just personally like the idea of the two existing in universe.
Same for her name. I'll leave that to her artist, if she deserves one.

I find it really funny the idea of another girl with her name who's everything she's not. Imagine that.
Like "This exchange student showed up and did everything I wanted in 2 seconds"
Of course, them sharing the same name is also not confirmed.
I love how her character can be taken into all kinds of directions, from wholesome to downright depressing or creepy.
Maybe she'll become more assertive, maybe her obsession is unhealthy, maybe she "Doesn't want to live" like you said. She's so different.

Her dynamic with the others is also very unique. Ichi-chan has to deal with someone who's treating her like how she treated Oguri Cap, Belicose possibly picking on her, Grisaria being her polar opposite.
No guarantee that they'll be friends either, anything could happen.
I'm very curious about how people will respond to her.