Not too expensive and Solent area are mutually exclusive IMHO! But you might look at Calshot Activities Centre or Drivers Wharf. DW is lift in lift out at I think 30 quid or so a time plus monthly storage.
Drivers Wharf gets my vote. As a berth holder it'S very reasonable craning costs, something like 15 quid+VAT for a lift, in or out on a 29' boat with a trailer. (Although I was accused of lying last time I said that) Plus there's a slipway just up the road if you want to launch/recover yourself, and a couple of nearby garages for refuelling. They do also crane at weekends, and it's generally a pretty friendly place. The only downside is the 2nd yard is shingle, not concrete.
Calshot generally seems to have a long waiting list to get in, plus no nearby fuel, no pontoon, and looks a bit exposed for launching if the weather is poor, and isn't very secure. It is very cheap and has great access to the solent though. I was also told there's a nasty ledge at the bottom of the slip and it is very steep, but I haven't launched there so can't confirm this.
Don't know but it might be worth a try speaking to Bill at Ocean Village and ask if there is any possibility of parking it on trailer in the North Car Park, then you've got the public slipway just around the corner under the Itchen Bridge.
If you approach him as a way of earning some extra revenue, he's a right nice chap normally, there are a number of Ribs parked up there but perhaps they are with bertholders I don't know.