Ok your WP Filesystem API Method is direct so file writing should work without any problems. Which type of Network / Multisite site do you have? subdirectory or subdomain?
Click the correct AutoMagic buttons for your Network / Multisite site type and activate Root Folder BulletProof Mode. If you see any error messages then post those error messages. If the problem is still occurring then go to the htaccess File Editor tab, click the Your Current Root htaccess File tab and post your root .htaccess file code.