IOS – How to restore a jailbroken iOS 6.1/iPhone 5

My iPhone 5 is running iOS 6.1, jailbroken. I've managed to lock myself out of the phone and don't know the passcode. Device is disabled for longer and longer periods of time in between incorrect passcode entries.

I've been backing up to iCloud and periodically locally to iTunes.

My objective: Ideally I would like to get back to my most recent iCloud (or local iTunes) backup, and stay jailbroken. My second most preferred choice is to get back to my most recent backup and then re-jailbreak. Worst case is bring the device to 6.1.2 (which is the latest version that there is a jailbreak for) and re-jailbreak and reinstall everything.

I am reading lots of conflicting and confusing things about how to do this, so I am not sure about the procedure.

My first attempt went like follows:

1. Attach iPhone to PC, iTunes sees it.
2. Download 6.1 firmware ipsw (because the latest version of iOS, 6.1.4, which iTunes tries to use, can't be jailbroken)
3. In iTunes, hold shift then click Restore iPhone, then I pick the 6.1 file

and I get the following much-mentioned but not much solved error:

The iPhone [device name] could not be restored because the firmware file is not compatible.

Doing this while the phone is in DFU or Recovery modes doesn't change the result.

I then tried the same steps, but this time selecting the 6.1.2 ipsw file. I get past the error above but after the file extracted and after talking to Apple's servers I get error 3194:

The iPhone [device name] could not be restored. An unknown error occurred (3194).

Lots of Internet activity for error 3194, all unsuccessful, eg. changing hosts file.

Recommendations? To reiterate, I want to accomplish what I wrote after "My objective" above; perhaps I'm going down the wrong road and this error message I am getting is irrelevant because this might not be the right path anyway.