how old is this refugium? i remember when i started mine, i had some die off in the beginning, and so did many others i've spoken to. i don't remember now if i had that striped problem, but i remember it not doing well.
meanwhile, plants thrive under the lower kelvin lamps- i use 6400k 23 watt energy saver bulbs like above.
i use a mix of a few different caulerpas as well, but i have to remove about 6" in a salt bucket a month.
( my lfs gives me 3 dollars a handfull for it)
i run my lights on a similar schedule as the display, since keeping it lit on an opposite schedule would cause my frag tank and display to lose it's night(they are all exposed and in the same room).
it's definitely not mandatory to run the lights 24/7, and i was under the impression that the only reason people did this was to cheat the caulerpa's out of a sexual event.
i don't see why you would have to run 24/7 for chaetomorpha...
sand beds/mineral muds/bare bottom, should not have an effect on the macroalgae's growth.
if the tank is brand new,i would be willing to bet, the lack of nutrients in the water is the problem.