OP, when we first moved here our border collie cross, unbeknownst to us, started howling whenever we had to go out without her. The move had unsettled her.
We had no clue it was happening until our neighbour (moved from detached to semi) came round a shouted at my dh. I turned out that the previous occupants had a dog that they'd left at home all day every day to bark and he was determined it wasn't going to happen again. At the time, we'd probably only left our girl for less than an hour, perhaps twice and his reaction seemed disproportionate - until we found out the back-story.
We did a lot of work with our girl to ensure we was settled and made sure she wasn't left at home alone until we were sure she could cope. We also kept the neighbours informed of what we were doing and apologised in advance if they heard her at all, but explained that we were doing everything in our power to sort the problem out. They were fine about it and relaxed as soon as they realised we were responsible dog owners. Their hearts must have sank when they saw us arriving with two dogs after their experiences with the previous occupants. We helped our girl over it and she was fine.
Fast forward 11 years, we've lived here peacefully and got along with those neighbours all that time - in fact we increased the number of dogs to 3 during that time and all was well, until last year when we lost the bc cross that had the howling problem. By then we'd already lost our oldest girl and there was only our Lurcher lad left. He grieved terribly and started to howl whenever I went out to do the school run. School is across the road and it takes me a maximum of 20 minutes to do the drop-off/pick-up, but at the time I was having to do it three times a day and again the neighbour went postal about it - this time leaving a rather rude post it note stuck to our front door while I was at the school, two weeks after we lost our old girl.
Again we went and spoke to them, explained we'd lost our other dog and he was grieving and that we didn't know he was howling, but thank you for letting us know, as now we can do something about it etc. We also reinforced that we wished they had told us before it got to the point where they are furious about it, especially as we couldn't possibly have known there was a problem if it only happened while we were out. I explained that I take my responsiblity as a dog owner very seriously and never want me having dogs to impact negatively on any one else, it's my choice to live with dogs, but that doesn't mean I have the right to allow them to annoy other people etc. Finally I reassured them that I would be doing everything in my power to retrain my boy to be able to cope when left alone and to ensure he didn't bother them if at all possible in the meantime.
They said they didn't know we'd lost our old girl and were sorry and said to let them know if they could do anything to help us with the separation anxiety. I spent the next couple of months staying at home, arranging dog-sitters when I couldn't be and very slowly building up the time I left my boy and doing lots of behavioural work until he learned to cope alone. There were, I think, two occasions when I had no choice but to leave him (important hospital appointment type situations) and I let them know in advance so they could choose to go out if they wanted to - but also put him on the opposite side of the house to minimise any disturbance. By that point though the neighbours said not to bother moving him and that they could easily cope with an isolated occasion (I still moved him though).
Once I went round and apologised, explained the situation and told them I was actively doing something about it they were fine, both times, but both times they had stewed for a while before coming and telling us there was a problem. I think this is probably what's going on with your neighbour and would recommend trying for a less confrontation chat. I am very shy by nature, hate confrontation and going to talk to my neighbours was my worst nightmare, but ultimately the best thing to do and definitely no worse than living with the constant anxiety that we might be annoying them enough for it to build up to another angry doorstep confrontation.
Incidentally, we then took on a rescue pup who - oh joy - also had SA due to his bad start (we didn't know when we took him on). So now I have been working with him for 8 months - making sure I spoke to my neighbours as soon as we got him and again, they've been fine.