This seems to be a hot topic now with many asking if Batman can be one or the other. I haven't really seen anyone ask if he can be both. Why can't he? We see other heroes do better than him.
SPOILERS AHEAD for the Button plot
In the past we've seen Roy Harper be both a hero and a father. We've also seen Booster Gold balancing fatherhood and his duties to protect all of time. Currently Clark seems to be surpassing Bruce in this area too. What we've been getting more often than not is Batman being a horrible parent when the plot demands for it and never really dealing with his actions. Let me be clear that I don't think Bruce has always been a bad father. Some writers show him struggling with both roles but ultimately being a loving dad. Others just show him being an ass or using his kids as soldiers. Jason and to a lesser extent Dick have questioned how he views them due to how he acts. (*1) Recently Damian has even started wondering if Bruce cares. He's referred to his father by his first name and expressed frustration with his parenting.
I'd personally be happy if Bruce could manage both.
*1 I have no idea what Bruce's relationship with Tim currently is. I'm not sure if Duke is adopted, etc. as I haven't been reading the main Batman titles so I don't know how their relationship is either. Same for Cass.