Using 28 AWG flat/ribbon cable for LEDs
I think that you will be fine.

You should expect a voltage drop of ~90mV/meter at 0.2A.

According to :
Fermi National Accelerator Laboratory
Ampacity Test of 28 AWG Ribbon Cables (1990)
lss.fnal.gov/archive/test-tm/1000/fermilab-tm-1657.pdf

28AWG Ribbon cable with parallel runs has been tested to 2.5A per conductor in free air, leveling off at about 40C.

Try it and find out for yourself.

Add a sprinkling of multiplexing to reduce the duty-cycle, and even more headroom is added.

Go for it.