I saw your edit that the wall may not be an issue. But I would still link the two gardens with insulated wire in the ground.That would be more than enough but the problem is going up and down the wall. I'm not sure we'd manage a way so that power would still go through- I know it doesn't work with my neighbour's netting.
Edited : I just had a look around to measure and the wall can be avoided on both sides. I think it could be doable with just one battery.
Part of your Partner’s objection is seeing the netting - and this way you can minimize it to just the planted areas.
My netting is green and I can barely see it if I stand back from it. The posts are grey and more visible, and I do hang some ‘caution’ signs on it in case someone is stupid enough to grab hold of it.