First of all you should get the Facebook IPA. I recommend you getting it from here. I recommend downloading the version v11 since it is the latest one for iOS 6 and it works really nicely, (at least on my iPhone 4s)
It might seem out of the ordinary to enable 2FA on your Facebook account when you wanna try logging in on iOS 6, but trust me.
To enable 2FA, go to the Meta Account Centre, then "Password and security" and "Two-Factor Authentication" I recommend using a authenticator app such as Google Auth or Authy, do the steps that you are given in the app and continue with the next step.
Enter your details on your legacy iDevice and (uh-oh) a pop-up should appear along the lines of "Confirm Identity" Click Open in Safari and you will be taken to a Facebook page with a text-box asking you to put a 2FA code*, to be fair you can completly ignore it and look at the notifications on your main phone, if you got a Facebook notification with a Warning sign, then click "Yes" If you've got the option. The browser should refresh on your older phone and successfully login into your account. *The browser might crash so keep trying until it doesn't, and if if you can't get it to work still then keep scrolling down when Safari opens (spam scroll to the down)