I think it's extremely interesting you've stated that families seem to be having trouble with the dog at the 3 month mark. This is the point in time when rescue dogs start to properly begin settling into their home - and also when pushing boundaries and problem behaviours tend to show. Note, I do not mean your dog is settling into the problem behaviour of biting.
Firstly, dogs never bite for no reason. Only a veterinarian or a qualified animal behaviourist could properly help you figure out what the reason is/was. Whilst I appreciate you've tried to be quite thorough in your post, the detail surrounding the actual events, and more importantly, the lead up to, is not nearly specific enough to begin to guess. When I say lead up to, I don't just mean that day. Look back over the past week or so & and have a really good think. A vet trip is an absolute must.
Secondly, get that dog off the couch and establish some boundaries. Once things have settled, and you've figured out what the problem is, you could potentially invite him onto the couch again from time to time, but you also need to have the 'down' command pretty established if so, and never allow him up unless invited into your space. I'm not diagnosing him with resource guarding, but if he does have an element of this, it is not always immediately obvious. Stress of feeling the need to protect a resource, like any stress, can build and build. The bite can come a week later, an hour later, later in the day. Not all dogs show clear signs until the bite. Or, the signs can be so subtle we don't notice until something happens and then we start to look closer (speaking from experience here).
Thirdly, you don't mention a crate. If you don't have one for him, do him a favour and get him one. Bare with me on this, precisely given his history with his crate. Get a cover for it (so it covers the back, sides and top). Make absolutely certain your child never approaches him whilst he's in it. At first thought, the idea of getting him a crate given his history of spending 12 hours a day in there for the first year of his life may make most people recoil in horror, but let's take a closer look. The early months of a dogs life is when they absorb the most. It is when they develop phobias, comforts, and socialise. What Bear learnt in his first year of life was that his crate was his home. For half the day, every single day. For a year. And for most dogs, whether they are crated for too long or not, that eventually brings a sense of security. If I'm correct in assuming from your post he doesn't have one, then he's gone and lost a big part of his life, his days, and his 'own space', completely. A bed, in an area of the house is not the same. He is still able to be approached in a bed. He can not really 'hide away' in a bed. There's not a single rescue dog I'd never offer an open crate to, at some point. Perhaps he's trying to establish where his place actually is, also. You need never shut the door of the crate. Or even actively encourage him to go in it. Pop it up, somewhere quiet, pop some items in that smell of him, leave the door open, with covers on, and leave it for a week or two.
I speak not as an expert, but as someone with two romanian rescue dogs - who have taught me more in that time than a lifetime of owning dogs. One whom had grown up in a kennel from at least 6-8 weeks old, for several years, barely socialised with people, and had never seen the inside of a house. The more affectionate dog, the more eager to please of the two, yet also the dog that bit me. It hasn't happened since, but I had to seriously get real with my knowledge of dogs, the psychology of traumatised dogs, phobias, lack of confidence and socialisation. The other a former street dog, very socialised with people, but came with a host of her issues, food and people guarding being just one.
In fact, I've just dug my crates out of the shed after over a year of them not being in the house. The dogs within 3 days, are calmer and more at ease already. The doors are open. They choose when they want peace. And we're a quiet 2 adult no child household.
Good luck, I wish you the best. I hope you and your daughter are okay. It can be quite jarring when the family pet bites you, I know firsthand. But with some factfinding and reading more into things, you then start to understand them better. Please do supervise your daughter whenever she is around the dog, at the moment, which I'm sure you already are.