I don't know if I would sweat the chloride that much. Since eggshell is already using it and hasn't seen any problems the calcium is doing more work than the chloride is causing problems.
Looking at what hits niveum, exul, concolor, godefroyae in the wild (i.e salt water) chlorides are obviously not a big deal to those species.
It's hard to generalize about peoples surface waters, but it's not uncommon to see 20-30ppm of chloride in most drinking waters (that growers typically use for watering orchids).
So dilute amounts of calcium chloride probably isn't going to put any more chloride into your plants than most growers get anyway from using drinking water.
However, I use calcium nitrate for my soluble calcium boost. Calcium sulfate (gypsum) is probably fine to, but has limited solubility (its a pain to dissolve). Bone meal (calcium phosphate) has been used in horticulture for years too, but its also not very soluble. Marilyn LeDeux (who I think is a fantastic Phrag grower) uses it routinely in her potting mixes.
You might also look at some of the Cal/Mag boosters from some other fertilizer brands. These will probably be some combination of calcium nitrate and mag sulfate (epsom salt). Recently I came across a cal/mag product (that I have not tried) for turf grass management that has calcium and magnesium chelated in organic sulfur materials (organo-sulfonates of some type). These may be really, good to try, but I have too many other things to play with already.