Abuse isn't just physical. A great deal of abusers go for emotional abuse and coercive control as their first choice or alongside physical abuse, because it's less obvious, easier to gaslight the victim about it being their fault, and so they get away with it more often. It's also not uncommon that even with only one case of obvious physical abuse, the victim can change their behaviour going forward to try to prevent a recurrence in future, so they succumbed to the mental side of the abuse because they know if they don't it can become physical. It's not necessarily a conscious decision but can be a subconscious thing, we are all programmed to try to survive. I'm seeing enough red flags in his behaviour and thought processes in this thread that I'm not going to discount her allegations.
Equally, he could just be a bit of a dickhead too - refusing to acknowledge his male privilege and the different ways men and women are socialised from birth. All this I'm hard done by, woe is me and I can't, I've got a career to think of bullshit.
From what he says she sounds a bit of a dickhead, but he's also not being entirely truthful, calling her full-time job part-time which is belittling and dismissive, refusing to acknowledge that teachers don't actually work 8.30-4pm, it's a lot longer than that. I'd be curious to know more about these "weekends partying in London" which sounds like she's either staying in hotels and clubbing, which is very expensive, or going to a lot of house parties and staying with friends. Given his general attitude shown here, it could well be that neither is true.
With abuse it's not uncommon to not recognise it for what it is, until much later. So acts of minor physical abuse becomes so normalised that the victim doesn't even realise what's happening is abuse and wouldn't describe their partner as physically abusive when asked, until months or years have passed and they can see the situation more clearly with better knowledge and are able to judge it accurately. With emotional abuse and coercive control it's even harder for the victim to see it for what it is. So alleging abuse against him, but leaving the child alone with him, isn't at all unusual. Many will also think that the abuse was only towards them not the children, so figure the children are safe (whether they're correct or not, it's a common way of thinking, as is not recognising abusive behaviour towards the children same as they didn't recognise it towards themselves). Then there's the knowledge that unless he's SA the kids or beating them to a pulp, almost everything is going to be written off as "parenting choices" anyway, so not much point trying to prevent him spending time with the children really, because the courts will usually give him some amount of unsupervised access no matter how awful a father the man may be. But, having said that, some people will allege abuse where there's none just to get legal aid.
OP isn't saying enough for me to form a solid opinion one way or another, which is fair enough because he's posted in legal, not in relationships, asking for specific help. So I'm very much on the fence about the abuse allegations.
Let's face it most men like to make out they're saints for doing anything and everything except actually parenting their children properly and carrying their fair share of the mental load domestically. When they split up, almost none of these men really want resident parent status, despite what they may say, because they want to prioritise their careers and can't do that so easily with a child permanently attached to their hip, so to speak. They prefer to leave that to the mothers, then whine about whatever the mother does to make her life better. At the end of the day, trying to prevent the mother living her life as she wishes to, under the guise of doing what's best for the child, is controlling whilst simultaneously deflecting and trying to garner sympathy. If the mother is that inadequate, fight for RP. If not, STFU and let her get on with her life. He can deal with his own feelings about the effect that her living her chosen life has on him by dealing with his feelings instead of trying to prevent those feelings occurring by controlling her actions.
As far as children getting moved around a lot, I can't get bothered by that. Plenty of children move around for various reasons, travellers and military children to name just two, they don't all grow up to be mentally destroyed by it. Ditto, children who see theirs fathers EOW.
The OP has choices, he's going to have to pick one. The fact that he doesn't like any of them is irrelevant. At the end of the day there's a reason most men have EOW, it generally works out to be the easiest option for everyone, including allowing them not to sacrifice their careers which is generally of great importance to them and EOW is usually achieved without too much fighting and legal costs.
Plenty of mothers aren't overjoyed at how having a child impacts their career or earning potential. But they see there's no other realistic option so they suck it up and get on with it without complaining.
He needs to stop bitching, stop wanting 50/50 access but only if he doesn't have to do the difficult bits - like compromising his career to prioritise his child - and only if he doesn't have to pay out to fight for it and only if he can have it by controlling where the mother lives because he'd prefer not to move around (which probably comes back to wanting to prioritise his career again). He admitted he let the relationship die due to prioritising his career. Now he's reaping the consequences of that decision. Although it may not have been a conscious decision but something that happened without him noticing, even that shows his thoughtlessness at the time and his focus on himself and his goals of wanting to be a provider. Not a bad thing to want, or wrong, just failed to realise that women primarily want love in most cases, over money. When she was post partum, recovering from birth and struggling with adjusting to motherhood, probably wasn't the best time to throw himself into studying for a new career alongside working full time and leaving no time for his relationship. Life is a learning curve.
To any men reading - as a general rule women will put up with a lot, especially if they've got children with you. For them to leave usually things have to be pretty bad and they often have to be feeling ignored, unloved, unfulfilled in life. If you don't want to break up with your partner, invest in your relationship first and foremost, that's my advice, because money, status and power won't cuddle you in the night.