I did an investigation and found an interesting thread.
https://www.dndbeyond.com/forums/du...ou-allow-upcasting-magic-initiate-spellsPost 9 quote (taken from Sage Advice compendium, which was officially linked by Jeremy Crawford):
If you have spell slots, can you use them to cast the 1st-level spell you learn with the Magic Initiate feat?
Yes, but only if the class you pick for the feat is one of your classes. For example, if you pick sorcerer and you are a sorcerer, the Spellcasting feature for that class tells you that you can use your spell slots to cast the sorcerer spells you know, so you can use your spell slots to cast the 1st-level sorcerer spell you learn from Magic Initiate. Similarly, if you are a wizard and pick that class for the feat, you learn a 1st-level wizard spell, which you could add to your spellbook and subsequently prepare.
In short, you must follow your character’s normal spellcasting rules, which determine whether you can expend spell slots on the 1st-level spell you learn from Magic Initiate.
So, according to that, 1) you can pick your own class as Magic Initiate feat and 2) the learned spell gets added to your spell list if you pick your own class as a Magic Initiate feat.
If it does not work that way in BG3, maybe Larian should fix that.