You can back up locally without backing up to Google drive just by turning off the option to back up to Google drive (change the frequency from "Daily" to "Never") and then tell it to do a backup. They actually take a daily local chat backup no matter how frequently you back up to Drive, but it's safer to explicitly take one.
I'd try copying the data to the new phone before installing WhatsApp, but after installing should be fine (though my instinct is to do it before activating). I know this is possible because my brother did it a few years back, but I don't know about guides (most web resources just tell you to use Drive, but that might just be because the instructions for doing that can be written in 1 line). The thing is to make sure you can find the folders with the media and the backups in them and copy the whole lot to the right place. On older phones/versions these used to be in internal storage/WhatsApp, but on my current phone they are in internal storage/Android/media/com.whatsapp/WhatsApp (which has 3 folders, Backups, Databases and Media). I'd copy the entire com.whatsapp folder and all of its contents. Hopefully with that stuff there you can just start WhatsApp then tell it to restore from the backup (or if you are really lucky and the databases are already in situ it might just load the messages).