Typeform provides a useful real world example of how to handle this problem, which is broadly in line with your third point:
Load the video (paused) and give the user an option to play the video?
On a desktop device the video plays automatically. On an iPad or and Android phone (the only additional devices I've tested) a still of the video is shown with a large play button overlaid. The effect is to present the video as the primary interaction avilable on the page. They've cleverly used a very arresting image (at the time of writing it's a dog licking the screen) which I presume is creating enough interest to generate plenty of play button clicks. The video itself opens full screen (i.e. it is not running in the background).
I think giving users the option to play the video is a good approach but as other posters have mentioned your final decision should be based largely on the importance of the video to your users. Typeform's is pretty compelling and nicely captures what the product does. In other words, it's worth watching both for the user and for the business. Compare this to New Relic, whose background videos on desktop are definitely just for 'atmosphere'. On mobile they just display a blank background with a clear, actionable message. The video doesn't add enough to the user experience to justify its inclusion on mobile. From a content strategy perspective you could ask why you would generate content that only adds value on a limited number of channels but that's probably a bit off topic for a UX discussion.