My only advice would require extra time, sorry. I have worked with my dogs - and it does take time - of when it is and is not proper to bark. If someone is in my driveway, then they bark and get praise and rewards. Otherwise, I am teaching them to be quite with people who wal byk. I do this by calling their name and the very second they look at me, I say "quite", and cup my hand over my mouth and give a reward. At first I had to do that over and over with someone walking by. Now, I might only have to say it once. I cup my hands over my mouth so that I can tap on the window, get their attention, and do the hand signal of "quite". It does take time for this training.
Maybe you could talk to your new neighbor, and tell her you are starting to train Daisy, but it will take time if she will be patient? The thing is, since you received a letter, I'm afraid she is not going to be very cooperative. You, and your dog, were in that house first! If it was a problem, then she shouldn't have moved in. I would first check to see what your city/county laws are. IMO, you have rights since you were already living there. Some areas do have nuisance noise laws, and some don't. The thing is, you don't want her getting really mad at you, if you can help it. You don't want to be afraid of what she might do when you aren't home. But, you,and your animals, have rights too, so it's a bit of a fine walk. I'm sorry, I do not know of any other good advice. I do know a friend who used a bark collar - against my strong wishes - and it did help Sometimes the dog would wear it and it was turned on, but sometimes it would be turned off---------BUT, never, NEVER use this when you are not around, if you decide to do it. I once had a dog who was wearing an electric collar, and it was being "set" off by other signals w/o me knowing it. Really awful on the dog. So, that would be my very, very last option, and I would Never, ever even put it on the dog unless I was home. Which is still going to take time to train.
edited to add: I just re-read your post, and I would NOT even consider a no-bark collar on a puppy, or a small dog. NO WAY. Sorry, I forgot about that when first reading. My friend has a 50 lb. adult dog and had the temperament to handle it. He is a very friendly, large dog, who just needed a little help with barking. I didn't like it going on him, but I can't even imagine one on a puppy or small dog. I really think they are easily abused, and must only be used under proper conditions (which, truthfully, I can't really help you with since I really don't have much experience with them - others on here might, but I don't.)
Has she just moved in? Could you try the approach of welcoming her to the neighborhood, introducing yourself properly, and if she brings up the barking - very calmly explain your situation. I guess what I'm saying is that I would try the "honey draws more bees" approach first. That's because once the bridges are burned, they don't get re-built. So, I don't think you lose anything in trying to befriend her first. Maybe she's lonely and just gotten cranky. Maybe she really needs a friend, and has forgotten how to get along with others. I would NOT give in to her request, but I would try my best to become her friend. I've lived next door to someone I had to go to court against and it is very hard living next to someone you can no longer be friends with. But, they left me with no option (They did have a vicious dog that attatcked me and they would not keep the dog out of my yard and a bunch of neighbors were just tired of the whole dog issue and the neighborhood took them to court in order to keep this particular dog confined). I really hated not being able to wave hello to my neighbors after that. I didn't have a choice, but I hated the aftermath of it all. I missed not being able to talk to them at all. So, I would try this approach first: forgive, befriend, forgive, befriend. With the holidays, maybe you could take a homecooked gift over and start from there. IF that doesn't work, then, I would consider the next options.(but the last one will burn all bridges).
edited to add this part: But, then just wait. Maybe she won't do anything else. Maybe she will let it drop. Thing is, you never really know how people are going to respond. Sometimes it might pay off to take a wait and see approach. It depends on her following actions. If she starts harrasing you, then you could consider the following:
Sometimes bullies (and I think she might be a "bully" since she wrote a letter, and didn't talk to you first) will back down when they are confronted. They act all puffy and big like they own the whole neighborhood and want things to run exactly their way without regard to other people's wishes. They think they deserve to have it all their way. But, I've seen where, when confronted, in the language they speak, they will, many times, back down. Since the letter is the language she speaks, I would get all the legal info I could about what rights you have, and send a letter back her way. It will cost a little money, but it would probably be really worth the money if it was sent on the letter head of a lawyer. Really. She is going to pay way more attention to a lawyer writing her that you have rights than you just stating it. Especially if you really tried befriending her first. But, at the very least, write her back stating your facts and your side of the story. Writing is good in that it leaves you time to be calmer, and speak more clearly.
A far as the old lady who throws stuff at your dog: Humfff, I
want to say to throw things back at her, but you could end up in court over that. So, in realizing that it is
your job to protect Daisy, then I would clearly, and plainly, and calmly tell that lady that if she ever, ever throws anything into your yard again, that you will report her for animal abuse and for whatever other court reasons you can come up with. She is vandalizing your property as well. I am not one to advocate going to court, but sometimes the threat of taking someone is enough to do the trick. I have no tolerance
whatsoever with anyone who throws things. And, this can make your dog more "barky" and defensive. I first would film her in secret and make sure you have the proof. Then I would let her know you have the proof, and she will end up in court if she ever does it again. You absolutely cannot allow her to continue throwing things at your animals. Period. She is wrong on every possible account. There is nothing about her approach that is right. So, stand up for Daisy. Get the evidence. Then let her know you have it.
I realize you are super busy, and all this just adds more time, energy, and hardship on you. It really is a bad situation, but the longer it all goes on, the harder it all will be to fix. Daisy has the right to be in your fenced in yard.