1) We know that since Featherine kills herself to alleviate boredom that she isn't omniscient. If she were, she would already know exactly what would happen with absolute certainty no matter what she did, so nothing she do could be a new experience and alleviate her boredom. Also, she has mastered several games in the past but have forgotten them. Omniscients don't forget, or have the need to master anything. Thus, it is actually possible for Batman to out-smart her. Batman could convince her not to kill him, that they once had a romantic relationship but she just forgot about it, or mind-f*** her in some way. Since he has full knowledge, he knows all about her personality and her past so with 10 years prep he can formulate some sort of convincing story. Once he mind-f***s her into liking him, he can just slowly manipulate her.
2) Featherine relies on that moon thing around her head to maintain her personality, memories, etc. Thus it is known that it is possible for outside objects to directly influence Featherine's mind. With 10 years prep and full knowledge Batman can create a device to simulate that moon thing and directly alter her memories. To power such a device, with 10 years prep he could gain items such as cosmic cubes or other infinite energy items. He could just make her his sex slave or something.
And no featherine is nowhere near LT level. LT doesn't need an outside object to maintain his memories.