Dont grow clematis montana as the weight will soon destroy their fence and the neighbours will likely become nastier, in fact nothing that needs support from the fence
An ivy hedge is not neighbourly and I agree with her, it needs to be dealt with, permanently
Honestly, I would prefer contoneaster franchetti but it is slow growing, makes a beautiful non-prickly berried hedge. The neighbours being nasty would warrant some leylandii, which I hate, you would need someone in to keep it trimmed to the height of the ugly cheap fence. Keep your side neatly trimmed and the other side will poke into their garden, through that horrible trellis. Sweet revenge