I think you should talk to the HOA first, just to find out your options. Because there may be a rule against doubling up on your side, and there's no point doing something that you'll have to undo later. Also, I'm not sure if the broken slats are visible, but if so, there's probably a rule requiring they be fixed. You could clarify both that and who is considered responsible for fixing it. (I know that, in this case, since your dogs damaged it, you're taking responsibility. But it would be helping to know who would be responsible if the dogs weren't a factor.)
Once you have all the information, you can go back to the neighbor. Explain that the fence needs to be fixed and the reasons why (dog, HOA rules, etc.). Explain the possible ways to fix it (doubling up on your side if that's possible, you fixing the fence yourself with access from their side, you giving them the boards so they can fix it, paying for a pro, etc.) . Ask them which solution they'd prefer, but make it clear that a solution needs to be found.