So I got some Phyco Pure Zooxanthella. Most of it I used to try if I can use it to reinfect my H. crispa, but that failed, likely due to the poor condition of the anemone.
The little rest of the Zooxanthellae solution I poured into a bottle with fresh, sterile saltwater. I added some Guillard's f/2 phytoplankton fertilizer and put it with strong aeration on my living room window.
Within 3 weeks the culture has gotten quite dense and under the microscope you can see good number of motile stages.
So even under home conditions you can culture Symbiodinium in a bottle.
I have a few bleached corals in my tank and will see if they can benefit from being set into a glass full with this solution.