I have an idea what if Marvel in the MCU made wolverine an Inhuman or at least something similar to Inhuman but his genes are changing mutanting(pretty much just make the X-Gene similar to the Inhuman gene but with modifications, but its still called the Inhuman gene(by mistake)) or make him like Scarlet Witch and Quicksilver the only people to survive experiments with Loki's scepter because of the X-Gene but its not stated
Have him not remember any of his past similar to the Ultimate comics, and X2 X-MEN United, but he has very vague horrible flashbacks of Weapon X but we don't need to see that its Weapon X yet(When Fox makes a deal with Marvel about it)
He should be looking for awnsers about his past and sees what S.H.I.E.L.D has been doing with Skye you know seeing her in the field and thinks she has powers maybe he can get awnsers about his past too, you know someone like him, so he joins the Agents of SHIELD after an encounter with her and becomes a regular like Skye or something.
Name: James Logan(Its Unconfimed) ( Its a name he never used in a the comics hey has used Jim Logan but or he could use a name of a Earth-928 agent his name his James Daly, just not Wolverine(since Fox owns the name or Agent Wolverine if they can use that then I say use it)
Codename: Agent Logan(since they can use the name Logan) or Agent James or Agent Wolverine(If Marvel can use the name)
"Inhuman" powers: Modified Vibranium claws(Marvel owns Vibranium not adamantium, but that is where the modified part comes in they could just say its modified with an harder than vibranium, but alien substance, its adamantium but not stated its adamantium so it could be a completely new metal), pretty much all the powers of 616 wolverine except he has part vibranium/ part alien metal claws(The alien part is adamantium but is not called it)
Personality: Exactly like 616 Wolverine
Actor play Agent Logan:
His X2 X-MEN UNITED COSTUME with elements of his current 616, and ultimate counterpart, but not too much for the fox lawyers to knock down their door.