As @WebElaine says, you can just change the post type directly in MySQL. Any post meta is mapped purely by post ID, so will all still be attached to the right posts.
Custom taxonomy terms should still be attached and if you register a taxonomy with the same internal name as your old theme used (look in its code for the register_taxonomy call) then you ought to see the terms magically reappear.
Similarly if you look in the post meta SQL table you ought to be able to find the meta key used for the video ID fairly easily and could either edit the database to use a meta key of your own choice or just carry on using the same key.
Essentially, my approach would be to use these steps as the starting point of your own small plugin for this portfolio post type, separating the functionality from the theme as you suggest you’d like to do for future robustness.