"A little research..." ... what research? Because honestly I know a lot of high level powerlifters, some who squat in the 700-900, who couldn't do a narrow grip overhead squat (or even a snatch-grip overhead squat) if their life depended on it.
Heck, I have a very good squat position, it has always been one of my best lifts. I also train a lot on the Olympic lifts (and even competed when I was younger) and I can't do a great narrow grip OHS.
The thing is that for most people, especially those with a muscular upper body, the close-grip overhead press will be limited by shoulder mobility. Not hip or even thoracic mobility. And shoulder mobility doesn't have much to do with your capacity to squat well. You need basic shoulder mobility to get the bar on your shoulders but that's pretty much it.
The pain you are having has NOTHING to do with improper hip or thoracic mobility and thus your incapacity to do narrow grip OHS is not in any way related to your squatting technique... and improving it will have no impact on the quality of your squats because what is limiting you in your OHS doesn't impact your back squat performance.
Zombie squats and front squats with a perfect torso/back posture will be much more related to a good looking back squat than narrow grip OHS.
I can see REGULAR/snatch grip OHS helping with proper squatting form but doing it with a narrow grip will not be of further help.
I'm not sure that your issue is a weak core though. PLENTY of elite Olympic lifters have the same hyper-lordotic posture you speak of. I'm not saying that it's good to have it but that it doesn't necessarily indicate a weak core. I could be due to overly tight hip flexors and squats... or weaker hamstrings ... etc.
Or it could be a postural issue due to bad habits (e.g. being seated a lot, always keeping your "core" loose when seated or standing).
People don't understand that postural correction is a 24/7 job. You can't expect to strengthen certain muscles and your posture will magically change.
If your body is used to always being in a certain position, that position/posture becomes the default/normal one. And the body will always try to get back to it since it now feels the most natural.
So if your problem is postural strengthening the weaker muscles improves your POTENTIAL to have a good posture but at first you need to change what your default posture is.
That means that at first you will have to voluntarily take and maintain the "optimal/new" posture. You will have to force yourself to stay in the proper posture. This means constantly thinking about it throughout the day, otherwise the body will simply revert back to its default settings.
That's why VERY few people are able to change their posture to a significant degree. Listen: the posture you have now has been programmed in your nervous system over many YEARS of CONSTANTLY being in that position. You can't expect to change that neural programing by doing a few exercises and thinking about your posture a few times during the day.
If you practice keeping the "new posture" and you are in that posture 1 hour a day. Assuming 10 hours of sleep, it still means that you are "practicing" your old/bad postural habits 13 hours per day. How can you change your neural programing if you are practicing the wrong thing 13 times more than the right thing???
When I was a M.Sc. student I worked with a chiropractor who was specialized in postural correction. He had his clients program their watch (or now smart phone) to beep every 30-45 minutes. The beep was a reminder to check if they were taking good postural habits (chest up, shoulders back, abdomen tight as if you are about to receive a punch, etc.).
Changing your posture is a full time job. Training and exercises is just a facilitator. There are no exercises that will fix your issue if it is postural. Only exercises that will make it easier to make the change.