Solent have no obligation to provide any service to aircraft outside their controlled airspace, whether it be outside the Control Zone or CTA. They are paid by the airport operator to provide service to aircraft arriving and departing Southampton. Due to having controlled airspace, they are obliged to provide a service to other aircraft crossing it.
As someone who is inside their controlled airspace, trying to get a word in on an overloaded frequency, with all and sundry outside, who thought that it might be a good idea to make a courtesy call, my advice is listen out on their frequency, if they are obviously busy and you are intending to stay outside, do just that and give them a wide berth, please don't call them. If you are transitting under their CTA, get their current QNH from their ATIS (113.350) and use it, not the regional QNH, to stay below the published base level, not at it.
When it comes to providing a service outside, Bournemouth are a LARS Unit, and are paid to provide such service by CAA. They primarily provide radar services outside controlled airspace, but can provide FIS to aircraft outside controlled airspace.
The primary provider of FIS is of course London on 124.75, but in theorey only. No way can they, with one frequency covering most of southern England, provide it to all comers.