Well this is all over the place.
Firstly you need to trim back the ivy on the fence your side. Ivy is rampant you won’t damage it.
Secondly leaves only kill the grass if left there for a long time. If you collect they have no impact on the grass.
Third I can’t really understand your grass, that doesn’t look like grass, it looks like a border, just dirt with broken kids toys and an old bin in it.
Fourth the conifers will be killing the grass, if the tree isn’t impacting her grass then you know it’s not impacting yours so something else is doing it, either leaving the leaves to rot or the conifers sucking all the moisture out.
Fifth, the sun moves round, there is no way she gets all the sunlight and you get none and she sits in the sun with the tree never casting shade at any point.
Six, leaf fall will mainly be on her grass, it’s not affected. So see point two.
Last if you wish to have the crown reduced then get a tree surgeon out, get a quote, if you can afford it then approach her politely with your request, explain it, explain why, show her what you want to do, and say you will pay fully.
If she doesn’t agree then ask if there is anything she would agree to.
Do not offer to contribute. If you can’t or won’t pay the whole thing then forget about it. You can’t ask her to do thr work and she pays. You want it, you pay.