I'm no expert - i've only been to Halong bay once for a day trip. It was lovely though. There are so many boats available, and some are not very big. So I must believe that it's possible to book a small boat with only a few staterooms. I can't make any comments about those providers. But I will observe that the boats you see on Halong Bay are not super plush and luxurious. But the day trip that I did was great fun, and the boat was quite adequate. The crew was great. And they provided a very nice, cooked on board lunch.
I realize that I have not directly answered your question. But part of my message is: It's a wonderful experience, and I would not get too wrapped up in the details of the boat.
On my next time at Halong bay, I'll definitely do an overnight.
If you are comfortable to play things loose and casual, you might be able to book when you are in hanoi, through the hotel concierge. That's what I did for the day cruise. But whether you could swing that for an overnight would depend upon low vs. high season. My booking also included the bus trip to/from Halong harbor from the Hanoi hotel.
And if you need a Hanoi hotel suggestion: Hanoi Essence Hotel in the old quarter. fantastic. and great breakfast.