MacBook Pro (Retina, 13-inch, Early 2015) Thunderbolt 2 port for small second display (ASUS MB169C+ Portable USB Monitor for eg)
No.  This will not work.  You will need to get a display with its own power supply.

Thunderbolt 1/2 doesn’t support power delivery.  Even though Thunderbolt 3 does, connecting an adapter doesn’t magically make power appear.

The (up-to) 9W of power is bus power - you can’t  assume that because it’s there, it can be utilized, especially to power a device.  Remember, Thunderbolt can support up to 6 devices, so it wouldn’t make sense that a single device allocate most or all of the bus power.