Ok, so there's a sleep regression at about that time, so you may find that in a week or so, this stops happening and normal service is resumed.
If not, you need to review their naps and bedtime, as early waking can be symptomatic of overtiredness (because yes, babies are that unhelpful 🙄)
4am is the witching hour, because by that time they've usually had enough sleep to feel rested (even if it's not actually quite enough) so they don't see why on earth you're trying to get them back to sleep.
Sleep won't be fully sorted until they can sleep independently of you - so no rocking / singing / patting etc. There are lots of methods of sleep training that don't involve controlled crying (we could never do it either - our son reacts like yours.) look at Pick up Put Down and gradual retreat. And there are others which other people might be able to suggest. You can also start by slowly reducing the input you're giving, so cuddling instead of rocking for example.
It's brutal! We hit a massive regression at 18 months with our previously amazing sleeping son, who woke at 4.30 every day for 6 weeks and only napped for 45 mins a day. I was in tears about it. And he just snapped out of it one day thankfully.
It honestly won't be forever. But 9 months is a decent time for some gentle sleep training if you want to do it. But you need to decide a method, and stick to it for a couple of weeks before you know whether or not it's working.
And drink coffee 
Your not alone!