There's the problem - OP cannot access the phone to back it up. It's effectively a dead phone with critical data on it.
The problem,
@jaarisnow is that the technicians have no idea what components to replace without full access to the entire device and the data stored on it. I don't know why a newer phone can't be used, but I assume you have researched it and have valid reasons.
From my limited perspective, then, I see two options for you... neither of which are good. You can either (a) have a phone repair technician work to recover your data, with full access to it; or (b) consider your data unrecoverable and wipe the device, with a permanent loss of data.
Personally, as a technician, I can assure you that cell phone technicians have seen it all: financial accounts, naked selfies, sex videos, and the most intimate, secret data people have. Unless your data is patently illegal, it's doubtful the technician will even blink at the content.
You can try to find a technician who shares your values and is, perhaps, willing to sign an NDA (Non-Disclosure Agreement) to keep your secrets secret; otherwise, you have very few options.
I don't like being the bearer of bad news: from my perspective, however, that's the best I can offer you.