I had not realised Southwold was your start point.
In that case Bradwell would probably make sense if you have decent weather and can get passed The Wallet and they do have Petrol (or did last year).
You will need to time your arrival to ensure there is enough water I would have said LW + 1.5 minimum based on our experience last year, although the marina claims access at most states of tide..........................
http://www.bradwellmarina.com/
Yes Chatham was on the Medway last time I looked
I had forgotten the need for easy access on and off your ship
There is also Gillingham Marina http://www.gillingham-marina.co.uk/mari ... _chart.asp
Nothing wrong with St Kat's and fully understand your motives and you may well get in with no wait but only right to point it out, forewarned is forearmed and all that.
But I think more to do at Chatham.
Just watch out for Rod in action as you head up the Thames