I know it more like this: if someone goes overboard with their hacienda in the outdoor area, the office, which otherwise does not suffer from a lack of work, has to take action. Only if the person involved then has a shooting club brother with a Schwibbschwager in the city council, the whole thing escalates and is expanded "on principle" to a land measure. Collusion always works (only) as long as some small faction thinks they are not getting enough crumbs from the cake. Then there is yelling and snitching to the superior authority.
But to the OP - I don’t quite understand two things:
His actual terrace is on the other side of the property
So why is he building a stage on your border – are you married to a MILF whose sunbathing could delight him?
And: where is your privacy screen problem? – if he does not get moving with his promised fence, then just put up a temporary privacy screen that is visually suboptimal enough on his side to motivate him to speed up. It doesn’t have to be something that needs to grow first – for all I care, bean poles with camouflage netting in between.
By the way, I find the suggestions for de-escalation worth considering: think of chain-link fence vs. pea shrub – in the end, both parties were the laughingstock of the nation on Stefan Raab, and even today – long after Lisa Loch – remembered.