We've not taken the boat there but we've taken the Beaver in there a number of times. In the summer your concerns are somewhat justified. The one long dock fills up and at the head of the inlet there is only one alternate anchorage and that is directly in front of the falls. The water in the entire inlet drops off very fast close to shore but there is a shelf in front of Chatterbox with room for one or two boats. You have to anchor very close in. But the theory-- which seems to work from our observation-- is that the current from the falls keeps the boat off the shoreline.
There is a small anchorage at McDonald Island just before you turn the corner into the head of the inlet. We've seen a boat in there every now and then but never more than one so I don't know how much anchoring room there really is there.
I don't know what the anchoring situation is like at Malibu itself. There's a strip of bottom about 40' deep along the shore by their docks but while we've seen boats at the docks I don't recall ever seeing anyone anchored there.
The uncertainty of getting a spot at the head of the bay is one reason we would not go there by boat in the summer. We do our longer cruises in September, anyway, so would consider going there then. There have been times in the plane we've flown in there only to find no space at the dock, but it's no big deal to simply turn around and fly out again.
There is a time limit on the dock. I think it's two days but you probably have that information already. But the time limit results in a pretty brisk turnover of boats.
It's a beautiful spot although compared to what's up the coast in BC and SE Alaska it's not anything particularly special. If you've never been to Chatterbox it's certainly worth visiting although it's a long plod back in there, even in the Beaver at 110 mph.