"I am being watched sleep" will be confusing to most and likely regarded as poor English. "I am being watched" sounds normal, but "sleep" seems out of place. If someone were to attempt to make sense of the sentence it could be unclear. When I first tried to make sense of it I understood it as: I embody ("I am being") sleep under surveillance ("watched sleep"). In other words I assigned "watched" as an adjective to "sleep" and took it to mean your personality is like watched sleep (whatever that means). So regardless of technical correctness, this phrasing should not be used because it will not make sense to most and will be likely misinterpreted by those who care to attempt it without sufficient context.
It may be better understood using asleep instead: "I am being watched alseep," but this is still quite odd.
Yes, watched can be used as an adjective (a watched pot never boils), but the problem in your case isn't "watched." If you remove it completely you have "I am being sleep." This doesn't make sense, you cannot BE sleep, right? So with or without the adjective "watched," you cannot use be + sleep as you have. If you want to use "watched" as an adjective here you can use: "I am getting watched sleep" as you mention, but that sounds terribly unnatural. The best option would be to just use "I am being watched while I sleep." or "I am being watched when I sleep."