I would say the smallest one that has actual Android TV would be the new Nvidia Shield tube - but I guess its not as small as you want.
Whatever you end up with, make sure it has external power. Typically pocket projectors don't supply sufficient juice on their USB connectors.
Also, prolly need male/female HDMI extension to facilitate solid connection.
Thi Mibox isn't a stick, but it's very slim. Might be an option?
Chromecast and casting from your mobile.
Yes that's what I am doing right now, but looking also for something portable platform based
Check this link for Google certified sticks :)
by the way - the tivo one looks promising!
Amazon fire stick 4k. It's almost an Android TV stick. It runs fire os, a modified Android firmware. It'll do the job
AirTV Mini
https://www.airtv.net/products/airtv_mini/
Tivo Stream 4K (not yet released)
https://www.tivo.com/comingsoon
closest you're gonna get to a 'stick'
I wouldn't consider the SHIELD TV tube one (less so than the above IMO - it's kinda its own thing), but that's an option too I suppose.