Sending sympathy as the mum of a 4 month old DS. The 4 month sleep regression is real! Your little one is already doing longer chunks than mine, who currently won't sleep any longer than 2 hours at a time. It's horrific. But I have a 2.5 year old DD so I know that it will all be ok in the end, and that makes it easier to cope with the second time around.
The bad news is, despite what others may tell you, there really isn't anything you can do. It isn't about 'self settling' (though that's great to master for wider good sleep practice), it's about them being able to link their sleep cycles - it just isn't the same skill, and unfortunately it isn't anything you can really teach. Don't bother upsetting her during the night by shush patting, just use the fastest and easiest method that maximises sleep for everyone. And formula and/or solids won't help either - it isn't about her waking hungry, it's because she's doing what we all do, and wake slightly after a sleep cycle. She just needs to realise that when she does wake, she doesn't need to freak out and cry out for you, she can in fact just close her eyes again and fall asleep. It's that linking of cycles that will resolve the issue.
The reason some say they just 'snap' out of it is because it literally is that simple - they just need to figure out that they can link sleep cycles together. As I say, plenty of people will claim they have the solution to teaching this but correlation does not mean causation and it's unlikely down to their parenting skills, it's just that their LO figured it out whilst they happened to be doing some particular method. Of course it helps if baby first falls asleep in their sleep space, and that they aren't too cold or too warm or too hungry when they do wake, but even if all of those needs are met, until they learn to link a cycle, they will still wake up after every cycle is finished (hence why I say you can't really do much). You are already doing all those things. Now it's unfortunately down to your DD to figure it out.
Some babies figure it out quicker than others. It took my DD about 6/7 weeks, and I did absolutely bugger all - kept feeding her at every wake, and slowly over the weeks, she started linking some cycles together and we got longer chunks. She then built on that and started linking more and more. And finally we came out the other side with much more stable sleep. Your little one is clearly already linking some in the middle of the night, which is great progress and means when her sleep pressure is at its highest, she can do it. The more she does this, the more she'll keep doing it
Desperately hoping DS doesn't take quite so long as DD, but we're 2 weeks in now and he isn't showing any signs of improvement yet. Little sod. It's fucking hard, especially as I then have a toddler to deal with come morning, but as I said, I take solace from knowing that this phase won't last (even if it feels like it when you're in the thick of it!)