She is being unreasonable, she can make you remove 'from' her fence - but she cannot stipulate that you do not pit anything 'near', that is ridiculous.
You can fix anything to her fence - this is called a right of support and you dont have one .... she may be worrying that the next thing is that her fence will be used as a back wall to a green house...
take the panel pins out- and have a free standing fence - you know pin it to sticky things that you put in your own ground. She can't for example say - 5 cm nuclear free zone on your side of the fence [ for the simple reason that THAT involves your space ]