Moving a one-click install

My client has had their site up for quite some time and upon trying to transfer their production WordPress setup to a staging on my own hosting (normal db/files backup), I realized they’re most likely on a one-click install.

Is there any way AT ALL to back up the files and shift the database over? I’ve been using Migrate DB which works perfectly fine but I can’t seem to get the plugin folder after trying to back up the files via other means.

Any ideas? I’m losing my mind!