To be clear, I don't think Bruce would have a problem with Steph fighting crime as Spoiler (thats mostly Tim's problem) or as Batgirl if Babs wanted it but I really don't think he'd want someone who was not "Family" operating out of the cave beneath his house. And despite the fact that Steph was a Robin I don't think Bruce thinks of her like that (the closest analogy would be Huntress operating out of his cave) and even though its Babs who took over, Bruce and Babs were not like that either in the sense of her being in the cave. Oracle hardly ever (once, twice?) operated out of the Batcave. She always had her own place. It's not like he didn't build a dozens of mini-caves throughout Gotham (including the one he gave Cass that's now vacant and the home-made version Jason made for himself) they could use instead and it's not like Babs can't just launder millions of dollars from some crime lord's bank accounts to buy/build a new lair for herself.
Dick closed the cave (and cleaned it out) for security reasons and out of respect for Bruce and literally one month later (publishing time) Babs and Steph move in? I don't think Bruce will like it when he comes back (and when he does they will definitely have to move out anyway if the Batgirl title is still going by then).