As I read it, Lily is not by default permanently added by clearing the main story. You must get the Silver Exchange Ticket (the grand prize from box 5 of the lottery) and exchange it in the shop. This means you need to farm approx 3200 - 4000 stockings.
While you could perhaps spend less in the lottery by resetting the boxes early, since this is a limited-reset lottery, it’s not generally recommended you do so unless you have cleared out everything “of value” from the boxes.
And FYI to the OP: you MUST make sure you get the Gilles de Rais dolls from boxes 1-4 of the lottery as these are Santa Lily’s ascension mats. If you do not get them all, she will be forever locked at a level below her cap (L80) at -10L for each one you miss. And as neither welfares nor their ascension items have made a reappearance yet in JP, it is unknown if they ever will.
Next, you will want to make sure you have 20 Buche de Noel items–these primarily come from the lottery, and since you get 5 per box 1-4, this will suffice. These allow you to buy the 4 extra copies of Lily from the shop to raise her to NP5. Considering that these are pretty much on your way to getting her permanent copy, there’s no reason to skip them unless you are desperately short on time, and might not have time to clear the lottery boxes. But remember–the Gilles de Rais dolls are your absolute priority–skipping any of them makes Lily just about useless. If you then have to skip NP levels, it’s not ideal, but an NP 1 Lily at L80 is still useful. An NP 5 Lily at L40 (or even L70) is not.
@Sizzle As a suggestion, while welfare servants and how to get them are pretty old hat now to most seasoned players, I think the event writeups should spell it out more explicitly for new players. I know I messed up myself with Kintoki and Irisviel in my first events, and didn’t get all their ascension items. Plus, I think it might even be useful to seasoned players (moreso for new events perhaps) who might be time-constrained and are looking to just grab thewelfare.