Hi everyone, OP here. Wanted to give the final installment on this because I promised it a while ago. Don't have tons of time for the forums but figured late is better than never. Hopefully it won't kick off another storm :)
Thanks for all of the input but unfortunately very little has changed and I haven't considered it worthwhile to go back to talk to the neighbor again or take any further external actions about it. After I spoke to her and gave her info about the best barking collars (plus some other suggestions), the dogs were silent for 2 weeks which I knew must have meant that they were somewhere else. I guessed she would do something like that, in case I went to speak to the other neighbors or anyone else. So relative peace and tranquility for 2 weeks and then the dogs came back and everything was more-or-less the same as it was. (During the time they were away, we did hear a couple of other dogs barking around the neighborhood a bit here and there, honestly probably also more than we were expecting when we consider the HOA rules, but not something that we would or could complain about and definitely nothing like the noise that her 4 dogs produce whenever they step foot into her backyard a few times per day.) So for anyone saying (or thinking) something like "oh of course the owners must not know that their dogs annoy people, otherwise of course they'd do something about it", well, I guess maybe you can see now that this wasn't the case.
It's pretty annoying to need to accept that we won't ever be able to leave our windows open. That certainly wasn't what we had planned when we decided to move here. As I said in my first post, we were REALLY careful to check about noise clauses in the HOA rules, especially relating to dogs barking, so we're really disappointed about this. It sucks that the HOA can just push it back onto us and make it our responsibility to push the police department to do something about it when a) we never expected to need to deal with the police department about such things, with the HOA rules being so clear on the issue, and 2) there is no police noise ordinance relating to barking dogs in this area anyway, so the cops just tell us there's nothing they can do unless the dogs are barking at night time or very early in the morning. Nevermind that they disturb our peace during the day - 4 dogs barking all at once is most certainly enough to interrupt our work, disturb our peaceful meals, wake up the baby, etc, even if it's sporadically during the day.
(And as I've said before, the dogs do bark INCESSANTLY by ANYONE'S definition, the issue is that they're inside most of the time and so they don't bother us usually. From our house, we can't hear them until they are actually let our into their backyard, which only happens a few times per day... but that's a few times per day too many when it's so consistent like that, every. single. day.)
We were originally planning to stay here for a long time but we've already started to consider moving. The dogs aren't ALL of the reason but they're definitely most of the reason (and we'd probably be more inclined to suck up the other problems, since no place is perfect, if we didn't need to deal with the dogs as well). My original question was asking whether we could force the HOA to reimburse us for at least some of the real estate costs etc if we we ended up moving soonish because of this. The truth is that we would never have moved here if we thought the HOA wouldn't enforce their own rules. Unfortunately, I guess it probably wouldn't be worth it to go through all that stress though. I mean, it's a lot of money, but it's also a lot of time and energy to fight for it.
Right now we're just trying to pretend that the dogs aren't an issue and just continue with our lives - well, as much as we can while we aren't able to fully enjoy our home with the windows open, and we need to have those big thermal noise-protecting curtains in the bedrooms, plus noise machines running whenever any of us are sleeping, etc.
One thing we've learned is that we're quite different to a lot of people out there. Aside from the fact that we would NEVER own pets without making sure that we take full responsiblity for them (24/7, not just when we feel like it), and we'd never do anything to even remotely disturb our neighbors, we seem to live a relatively peaceful life. When we went to talk to the neighbor, we thought the noise level in her house was incredible.
Even without considering the 4 dogs barking constantly inside, she also had a TV blasting, etc. It's hard for us to understand such chaotic homes, even though now we do notice them a lot. No wonder she can't empathize with us - she probably can't think at all in that sort of environment. We have a TV in a room separate from the main living area, just for watching occasional movies (when we specifically choose to watch one, e.g. a couple of times per month), but no cable or anything. Our house is silent except when we are talking/laughing/playing or occasionally listening to some music (at a respectable volume, i.e. not audible outside the home!) while we're doing some cleaning or cooking. So of course we're more likely to notice annoying noise coming in from outside the house than most people.
The one good thing to come out of this is that the neighbor does walk the dogs every few days now. It's still not enough exercise for them, but I guess something is better than nothing. At least their lives are a bit better now, even if ours aren't.
Anyway, I'm not looking for any advice or anything, and sorry about the long reply, just wanted to give an update because I said I would, in case anyone was interested.
Thanks again everyone for the advice and empathy, to anyone who has dogs, please look after them, and to anyone who finds this thread later because you're having trouble with neighbors' barking dogs, I hope you have better luck than we did! :)