Researching a bit further, Lola ran away from Totto Land when Pudding was a little girl. So, we know Pudding is currently 16, so she was 2 when Harald died. And it was 5 years ago when Lola had her shadow taken by Moria, a year after Loki was defeated by Shanks, where Pudding would be 11.
This photo from the wiki would tell me she was a bit younger then 11:
https://static.wikia.nocookie.net/o...ng_Lola.png/revision/latest?cb=20171112035442
So, yes, whoever exactly reached out to Big Mom, it absolutely was not when Harald was king. It had to have been set up afterwards, either by Loki trying to trick Big Mom....or Jarul swallowing his pride and trying to bolster Elbaf's security with her crew. Which would mean in that case, it was also a fake Loki.