How can I enable HTML output for checkbox options?
For starters, if you're using an admin theme like Seven for your administration pages, nothing on those pages will ever be affected by another theme's code. If you need to write code that applies across any theme, then you should be putting that in a module.

Inline styles are an attack vector for XSS, which is why they're generally not allowed in any kind of markup that is the result of user input. That includes the label text for a form element, such as a checkbox: the label is constructed by theme_form_element_label(), which sanitizes the text with filter_xss_admin(). That's absolutely the correct thing to do.

The proper way to do this is to put your styles in an external spreadsheet, as you've mentioned. Consider whether you could implement your code as a module instead.