Is there a non-stabilized option that will add FC without CYA?
Absolutely. With your low calcium levels, you can use granular Cal Hypo on a daily basis.

Your test results appear very precise...that's good. I would suggest the use of Cal Hypo and test the Ca level about once per week....it'll slowly climb. There's a pretty wide range of acceptable Ca limits (somewhere around none to maybe 500) so, once the Ca tested about 300, I would stop and switch back to bleach. That may be all Summer in your pool.

Granular Cal Hypo is a good shock source as well.