How can I prevent "Reset network settings" on my iPhone from affecting my Laptop?

Check to seen if your passwords still exist under Safari > Preferences > Passwords on your Mac. Maybe they are still there.

If this doesn't work, try the following: Unplug your router's ethernet cable (so that iCloud cannot sync with the server) and try to restore your from Time Machine. This should revert your Key Chain. I recommend disconnecting your router because if you just turn off your wifi, it will probably turn back on after the restore is done and sync with iCloud again.

I don't know how to prevent it from happening, but I fixed it last time by getting a copy of my keychain via TimeMachine and reimporting them.