If you can't change permissions or your hosting provider, you had better change your hosting provider. I am not into recommending shared hosting providers but if this is the only choice you can go with A2hosting or Bluehost. If you can go VPS try: Linode, Digital Ocean, and Vultr. As long as you have the knowledge and the will to set up everything by yourself (server security, software installation, and maintenance).
Wpengine seems good but only for Wordpress sites.