I would look at separating the power isolation
from the signal isolation. The low current of the
small "solar cell" is the limitation (incorporating
also, the limits of LED power and overall
efficiency).
If (say) you left DC power to the LED on all the
time but used it to charge a capacitor bank,
and used some separate opto path or "split"
the opto path by LC, CR filters such that a
modulation could be superimposed on the
base light, and added a MOSFET gate driver
driven by the detector for said modulation,
you would get fast turnon / turnoff. Of course
rep rate will still be hostage to basic current-
throw in the opto power path, and you'll pay
for all the "off" interval with LED current that
after the cap bank is charged, contributes
nothing but heat.