Hmm… So, let’s layout what we know.
Our web servers run on Apache. By default Apache requires our web directory to have a www-data set as the group for all of the files and directories. This paired with Permission 775 to allow both Read/Write access for www-data. We have already confirmed that this is correct.
Magento, requires it to set the owner as magento, again with the Read/Write privileges. This is because Magento also needs to make file modifications… Of which are executed by the cron-jobs we have setup as the magento user. We also know that this is confirmed to be correct with your setup.
So, amending the permissions to 775 allowed Magento/Apache to make the necessary changes to the wordpress directory during the Magento Module part of the installation. 
Test 1
But now for whatever reason, we’re on the second hurdle, where Wordpress is complaining about privileges. That only leads me to 1 theory right now: More Files/Directories have been added to the wp/ directory (maybe by magento during the last step), therefore you need to run the 775 commands again to the entire wp/ directory.
Test 2
If this does’t work then you can attempt to run an experiment… To set everything in the wp/ directory to 777 (never recommended), restart the install process. If it works then identify what those Files/Directories where created as (owner:group). For some crazy reason, it might turn out that Wordpress is trying to write files as bob instead of magento or www-data - which would explain why 775 doesn’t work. Unlikely, but we’ll see.