On the Studland side, I think Shell Bay is the only place you can launch a boat with an engine and its expensive and not sure they do ad hoc launches.
The National Trust Beach at Knoll Beach has a slip but it can only be used for dinghies and non motorised boats
Nowhere else on this beach really has car/trailer access.
There are a number of very small slips/ramps on Banks Road the other side of the chain ferry on the Sandbanks peninsular which are only suitable for hand launching (and at high tide!) but are really for the windsurfers & kayaks - you may be able to get a Bombard launched on one of these with difficulty - if you can find adjacent parking.
Otherwise as Whisper says Baiter is your best bet. There is also a public slip in Swanage.
For such a large harbour there are pathetically few casual public launching spots!