How to Change Price Set Field Type without deleting and re-adding
My first answer is: yes you can do this in the backend! I just confirmed it: in the civicrm_price_field table find your Price Field ID (row) and then change the html_type from: Radio to: CheckBox I just did this on a dev site and my price/field?action=preview looks just fine - no errors/notices.

My responsible answer is: first make an extra backup - next make a clone of your project and make the edit to the html_type - check your contribution page(s) where you offer this price_set - run any report you would normally run to check to see that everything still adds up [IF you've already had users selecting these price_set fields so you have data in them already].