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.