Hi Geordie,
I fear my comedy vernacular may have been misunderstood.
I guess my base message is that you seem like a very decent and reasonable person but it seems like the person you are dealing with is completely unreasonable, and perhaps is actually suffering some kind of mental illness.
If this is the case then all your attempts to compromise will inevitably come to nought, as a decent person you will fret and worry when instead you should be enjoying a perfectly reasonable and normal hobby in your garden outside your house.
My comedic thrust is that unless you can dismiss her complaint as utterly ridiculous then you risk your enjoyment of the hobby we all love being compromised. That isn't fair, it is passive aggressive bullying on her part, I hate bullies and tend to respond in the strongest terms.
As many more sensible, and less colourful posters have suggested you can seek compromise but do it from a position of knowledge and confidence that you are doing nothing wrong and that you should ultimately stand for your right to enjoy a barbecue in your own garden without threat of accusation or guilt.
Seriously yours
Tiny