Thanks - yes the marina was the answer - very happy to let me park car and trailer there for a frankly very small amount of money.
I knew of the slip by reputation but this was first time using it. So in fact the trailer spaces (8 or 10?) look pretty good, dedicated tarmaced bays off the main road. I'd have been OK leaving a trailer there, but the car was the problem.
I did call Dover Council parking dept, turned out the number went through to a general "customer service" dept, who sounded well-meaning but were useless. On explaining I wanted to use the public slip, launch Saturday am and recover Sunday pm, they knew nothing about any dedicated trailer spaces but rang the parking dept, who suggested we use one of the coach bays by the sailing center. When asked if I could get a 36 hour ticket, they said no, max is 9 hours, you just have to buy a new one Sunday morning. But I wasn't going to be there Sunday morning, that's the whole point... well tough then. And when I got there yesterday the coach bays bore very clear signs prohibiting both cars AND trailers. So that was useless advice....
Launching was easy - what no-one mentions are cobbled slips either side of the concrete one,and much wider (but shorter). We happened to be launching HW+1, so using the cobbled slip (which was dry) was easy, despite signs warning the cobbles could be slipppery when wet. Looked to me was though the cobbled slips would be useable HW+-3'ish? ( a local will know better)
Recovery was at low tide, so had to be the concrete slip. This took a number of attempts to get lined up (and you need a spotter on the sropoff side) but once we were, not too bad. But the boat was in a metre of water or lless for quite a while on the way in (spent some time toight flushing sand/mud from the cooling system). Once on the trailer, driving up the rather weedy slip was OK
So all in, a tricky slip at lower tides, but quite useable. And the marina was the solution to the parking problem, Dover Council clearly were't...