Why don’t you try to approach this in the reverse way. On non-mobile devices the video is set to auto play but by default the play requires a user action.
Also, if the decision is based upon the fact that you don’t want video to over run user accounts then it’s not mobile that you should be testing for but the connection type/speed.
I had a 4 Gig monthly limit on my phone sim so video isn’t a problem for me, but the time it takes to load the video just for a background isn’t worth my while as an image will do just as good a job.
On the flip side, my wife has a 500mb limit on her phone so bandwidth and download is an issue, but when she’s out and about she’s likely to tether the mobile to the laptop and browse that way.
Fortunately YouTube does a lot of the bandwidth stuff themselves so that would be a help.
For these approaches I usually avoid background video (personally don’t like them) but if a client demands it then I say for anything 800px and larger they get the video, smaller they get the image (which is also the default).