I can understand you desire not to annoy new neighbours but no, you aren't being unreasonable. If the fence definite,y belongs to your neighbour and the agent isn't just wriggling out of a repair I agree, the police seems heavy handed and I'm not sure they would respond. There could be all sorts of other official channels and if so I am sure someone will be along to exp,aim. I'm wondering if there is there any way you, or someone on your behalf, could fix something stronger and more permanent on your side of the fence. Should be possible, even if you need to put in two fence posts in your garden to do it. It is a shame when you like the house and had hoped to stay for a long while. Neighbours can never be predicted sadly. I think I am a little worried for you As they don't sound the most friendly. Have you ever chatted with them? I hope it all works out for you and someone else can be more helpful.