If it were me, I'd go for either Calshot at bottom of Southampton Water, or Camber Dock at Portsmouth. Both highlighted on the attached piccy.
Camber Dock is great, but bring enough pound coins for the parking meter, I think about £8, but it may have gone up. Also, £2ish for the slip barrier, it it has been closed. It's a really good steep slip with somewhere to tie the boat after launch and before recovery.
I've never lauched at Calshot, but at least the one on the end of the spit is a good steep and wide slip. £14.50 per boat up to 6m, including parking for car and trailer, according to baot launch.
Calshot has the bonus of being nearer the New Forest, so accessible to good/nice campsites. You can also camp at Calshot, but it might only be for members.
Alternatively, depending on tides, you could camp at Island Harbour up the Medina. You lock in and out of the marina when enough water, but it's quite nice. May prove a bit difficult taking all the gear in a SR though...
Am back that weekend, so may well join you, not sure if it will be in the SR though