I believe that you can SR before receiving it from the deliveryman to reroll the nature / HP type, but I’m not sure.
KRLW890 is right. If you save before you get the gift from the delivery man, you can soft reset and get different IVs and nature.
But since you have to get the right IVs and the right nature, it will take a while. The IVs needed for Hidden Power Ice are 31 in HP and Attack, and I think mystery gift Pokémon are assigned 31 IVs in three random stats, so you actually have a pretty good chance of getting it.
Source: I used this method to get a modest Dialga from the 2018 legendary year event.
This didn’t exactly answer your question, but I hope it helped.