Without root SSH access, you are limited to the features of your control panel. Perhaps it has some sort of API to upload the certificates without manual intervention, perhaps not. You didn't specify your control panel and also your hosting provider can probably disable such an API if it even existed.
Tell your hosting provider you're not satisfied with it's current policy of SSL (but they probably won't care, as it is a method of income for them) and tell them you'll switch to another hosting provider which does support free and automated Let's Encrypt certificates. You can find such a hosting provider in Web Hosting who support Let's Encrypt