I have had this idea for a story going on in my head for a while now on how Ori got his apprenticeship, the only problem is I can't write but maybe one day I'll fully flesh it out into a story.
In my head when Ori discovered his craft (scribe work) he didn't immediately tell Dori about it. This was because Dori's craft was tailoring and clothes-making, which Ori knew was a very costly trade due to all the materials needed and they most likely wouldn't be able to afford the supplies for both. Ori also knew that if he told Dori about it Dori would most likely at least partially put his own craft on the back burner in order for Ori to be able to pursue his own craft. (Especially if that meant a higher education for his younger brother)
This leads to a panicked Ori lying to Dori when asked what he wants to take as an apprenticeship, with Ori saying he wants to study combat. ( In my head due to the few number of children in ered luin every child is required to be taught how to fight anyways but it would just be more extensive) But at least he gets to hangout with fili and kili all day.
But when the time comes to begin the training he often finds himself sneaking off to the library to watch balin and the other scribes work.
Eventually balin realizes that Ori is just there to skip out on combat training he's actually interested in the craft (a rare case among young dwarrow) and lets Ori help around the library carrying books and scrolls for the elder scribes, and sometimes being asked to help copy documents and scriptures.
Eventually Dori has to come pick Ori up early (I imagine he has to go bail nori out of jail again) and is unable to find Ori until either fili and kili accidentally rat him out saying smth like "idk where he is but he's definitely not in the library" or turns out the guard -probably dwalin- knew he was sneaking off to the library this whole time.
Once Dori tracks down Ori to the library he finds him sitting in a corner reading a history book talking about the first age since he had already finished all his tasks for the day
Right before Ori and Dori are about to leave the library (after Dori quietly scolds Ori because "you shouldn't run off like that! I had no idea where you were. What if you were kidnapped?) balin talks to Dori about letting balin personally tutor Ori because it's hard to find someone with that level of appreciation for the craft -especially when most young dwarrow would rather study combat or smithing compared to scribe work.
long story short Ori gets to apprentice under balin teachings :D
Also this provides a opportunity for Dori/balin.