If I simply copy the multibit.wallet file over my current one would it work?
I agree with the other posters here that this should work, assuming you back up anything before overwriting it. I'm concerned that in the screenshot you provided, there was a noticeable change in file sizes between the backup files. As long as you get back all of the addresses/keys you need, this seems the easiest & safest option.
Or if I have access to the private key backups could I just import them into a new client and bypass the whole password process?
The private key backups (at least the ones that MultiBit creates automatically) are encrypted with the same password(s) as the wallet files...
If restoring a wallet file doesn't work, you can try to look for password typos with this:
btcrecover (full disclosure: I wrote that...),
quick start is here.
Good luck...