SOLUTION!!!
The problem is a max_input_vars variable on the server side. Most servers have this variable set to 1000 but this is not enough for larger pages – thus, it starts cutting off all edits made to a single page because it considers the page to be too long.
The fix was simple:
Create a file named .user.ini with content max_input_vars = 10000;
Upload the file to the public_html/ folder via FTP.
Thanks for your input Nisha and wptit. Hope this solution helps others having the same issue with BeTheme.