Crawford, the rules sage for 5e, has said the intention for the Magic Initiate feat is that one would be able to use their own spells slots to cast the 1st level spell that is gained by the feat. But, the 5e feat does not explicitly say this, so Larian implemented it in a fashion where you can only cast the spell once by Long rest. So, using the feat to add to one's 'known spells', to one's spellbook, or to one's spell list isn't really an option in BG3.
