Can I add styles to footer with $wp_styles->add_data?

I don’t know what your exact reason is for this, but you should scrap the idea of moving styles to the footer. If you though of a gain in speed, you might gain a unnoticable amount, if any, but that will be at the cost of other bigger things.

Styles should always be added inside the <head></head> tag. The reason is that <style> tags outside the head tags are invalid HTML. That is why you don’t have that option to load styles in the footer with wp_enqueue_style and wp_register_style like you have with scripts.

I would seriously rethink this whole idea.

EDIT

As a sidenote from a comment to this answer from @G.M.

wp_enqueue_style add <link> tag, not <style> one. However, <link> tag is not allowed outside <head> too

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