Could try Elkins Boatyard as well. Space is limited in the area with most likely visitors put of by the shallow entrance. I was going to suggest the sailing club but you having a motorboat would possibly have no luck unless midweek out of season.
I've sailed cats off the beach on the way in.
Definately stick to the marked channel and look for a good tide.
We used to sail ashore onto the beach and just outside the marked channel the sea bed is VERY hard - rock from memory - also flows fast through the gap.
If you just want to get alongside for a trip then you options are a bit limited. From bitter experience I would not recommend going along side on the pontoons around Tuckton bridge without permission as the berth holders can be quite possessive. Although not allowed I have seen boats tied up to Mudeford Quay. As suggested earlier your best bet may be to give Rossiters or Elkins a ring during working hours, they may have a berth available temporarily.
As others have said, there are walk ashore pontoons up the river by Rossiters and Elkins, but they're not visitors' pontoons in the usual sense. You could try calling Rossiters - if all else fails, then they can probably give you a non-pontoon alongside berth, either in the pool or next to the fuel berth on the river. Tidal range up there is only a couple of feet, so not having a floating pontoon is not much of a problem, but just watch the depth in the Run and channel.
The only rocky bits to watch out for are a ledge extending out from the middle of Mudeford Beach, and the extension of the breakwater at Hengistbury Head. The Run and the harbour are mostly very soft mud with some shingle around the bar.