Can I hack into my own disabled iPhone, without deleting the data?

My iPhone is currently stuck on a screen that says "iPhone is disabled. Connect to iTunes." When I connect to iTunes, my phone doesn't show up as a device. (I've tried on my own computer as well as a few others.) My phone is just frozen like this, all I can do is shut it down. If there's a way to attach an image here and i figure it out, then I'll show you what it looks like.
The Apple store can't do anything but reset the phone to factory settings. It hasn't been backing up because I haven't had enough storage, so if I do that, I'll lose some pictures I'd rather not lose.
What I'm wondering is if there's anything I can do that's maybe too shady for the Apple store to do, like hack into my own phone without erasing the data, so I can download everything I want to first. I downloaded software that said I could do this (Dr. Fone), however it later contradicted itself and said that it would erase all my data, so I need to try something else.

Not possible, even a jailbreak can't do that. Your only option is to reset it.

In future, I recommend doing backups on your computer when your iCloud storage gets full.

Yeah there's nothing you can do about it. The only somewhat feasible way would be to go through a company like Cellebrite which sells devices that can make backups of phones. However, they only work with law enforcement and other government agencies.

That's why backups are critical!

If in your computer, you cash get into your i Cloud and transfer a lot of stuff into the HD of your computer, this will free up Cloud storage. A very thin possibility exists that the Apple store people can connect your phone to your cloud account and move your pictures there. If yuh have another phone - even old and unused - that you can also connect to your u Tubes account, yuh might be able to salvage pix thru Photostream. Maybe even move Cloud stuff, leaving storage for photos in compromised phone. These are SLIM chances - but we have rescued a few things with such tricks - we always keep our previous "no service" found Sund to use like m3 players I sleep with mine using earbuds for podcasts) and one more way to get dates by the services staff things like photos.

Try different cables, in case the one you have used is faulty or fake.

It likely must be an apple-approved one (not necessarily made by apple, but a maker thay have licenced).

The Lightning plug has complex electronics in it and if the phone can't identify whatever is connected to it, it will not get any further in the connection process.