Can I shoot a bowfishing arrow that has the steel cable hat attached to the line thru a recurve?

Absolutely.

Cables work just as well now as they used to.  The Muzzy slider is nicer, IMO, but I wouldn't hesitate to shoot a cabled arrow if there was a fish in front of me.  Just make sure you have the cable away from your hand and bow.

I shot cabled arrows for several years until I switched to AMS Safety Slides. Cables work just fine so long as you set them up for maximum clearance.