How to get my passwords out of keychain access (and Firefox) from my CCC backup without a Mac?

My MacBook Pro (running 10.11.6) has a swollen battery, so I shut it down promptly and am sending it in for service. I have an up-to-date backup (a bootable backup) made with CarbonCopyCloner. The servicing of the Mac will take a while (maybe a week or so).

In the meantime, a lot of my passwords are unknown to me. They’re mostly auto-generated fully random passwords, saved in either Keychain Access or in Firefox’s password-saving feature (or sometimes both). Obviously I know my login password and the passwords to unlock my keychains (though I often just use the default login keychain), but I don’t know most of the other passwords.

Since I do have email access through other devices, I could reset various passwords. But that would be a lot of hassle, and then when I get my Mac back I would have to update the keychains and Firefox.

I have other computers available running Linux (Ubuntu) and Windows (both 7 and 10) to which I can connect my CarbonCopyCloner backup. I also have an older MacBook Pro from 2010 that I almost never turn on; I think it’s running Mac OS Lion (10.7) but it might be Maverick (10.9?). I’d rather not use that one if I don’t have to.

How can I get my passwords out of the keychain(s) and Firefox data from within my backup?

(For completeness, I actually do still have my laptop with the swollen battery since they’re sending me a box in a couple days to use for shipping it. But I’d really rather not turn it on given the swollen battery and potential for damage, plus I’m curious about this question for its own sake.)