I love this! This actually inspired me to write it! (If I finish it is another story)
I imagine that Danny just starts hanging out in the Watchtower every once in a while, probably no less then once a month. The JL don’t know what to do because they don’t want to offend the most powerful being they have ever encountered. John tells them to make a offering table with different food and drink, and promptly refuses to go on the Watchtower whenever Phantom is there.
Diana tells the JL that even Zeus feels weak when compared to Phantom. This of course sends Bruce into contingency plan mode, except he can’t think of anything. So, whenever he can, Bruce asks Danny questions to try and figure out a weakness. He either gets stories of Danny fighting gods told in such a way it makes him sound absolutely terrifying, or the most cryptic and vaguely threatening answers he has ever heard.
One day Barry asks Cap why he never called his dad sooner, and Billy tells him it’s because his dad was asleep and he didn’t want to wake him up unless he absolutely had too. Funny enough, Danny has gotten into the habit of not sleeping for around 1000 years and then sleeping for 100; consequently, he was actually woken up a few years early by the summoning.
This fact combined with some of Danny’s answers (like the fact he was around for the creation of the source) leads the League to conclude that Phantom is actually Chaos. (For anyone that doesn’t know Chaos is the Greek creator)  The theory is stronger when you account for the Eldridge form Danny can take. 
Meanwhile Billy is being combat trained by Fright Knight, because Danny is scared he’ll accidentally hurt him. Danny mostly teaches him magic, and does target practice.
Cap gets called to the Watchtower as soon as he gets back from one of these sessions, and the Justice League is terrified when they see him all scuffed up. They ask him what happened and he just replies:
“Don’t worry, I was training with my dad’s royal guard captain, dad still doesn’t want to spar with me because I’m so young. But he said as soon as I can defeat the captain in one on one combat I can finally practice with him! But that probably won’t happen until I’m a few million years old… Fright still hasn’t stopped babying me…”
This thoroughly freaks everyone out because they have seen this man fight Gods, and come back with not even a scratch. And if this was whoever this ‘Fright’ is babying the Captain, how powerful are they? More importantly, Phantom won’t spar with his son because he is so much more powerful than who is currently the most powerful member of the Justice League.
Needless to say, Bruce is very quickly turning gray from this ordeal.