How do I change the permalink structure without getting 404s?

If you can’t modify .htaccess then chances are WordPress can’t either. Save your permalinks again and check if it tells you it couldn’t save your htaccess file. That’s your problem if so.

Bigger question: why are you using a host that won’t let you modify .htaccess? That’s like renting an apartment from a landlord that won’t let you use the bathroom. You’re paying them for space on that server and you should be allowed to use it in any legal way you see fit. My advice is that you change hosting providers.