If you intend to go down the small claims route, you would be well advised to write to the dealer again. Clearly lay out what you expect and allow them 14 days to pay. Tell them that if you have not had a settlement after 14 days, you will make a claim through the Small Claims Court.
If, after 14 days, you don't get the settlement, make the claim using Money Claim Online. IIRC this will cost quite a small amount. I would not instruct a solicitor at this stage, but you could seek a free hour's consultation to satisfy yourself that you are likely to win. You should spend no money on independent reports at this stage as the courts are likely to appoint their own independent experts if necessary. You and the dealer will split the costs for this. If you instruct your own witness now, the other side can choose to ignore the evidence, and you will have to foot the bill.
It is unlikely that the dealer will want to go to court. If he does contest it (counter-claim) expect it to drag on for a *long* time. If they do not provide any supporting evidence, you will win by default. It may not be necessary to use a Solicitor as long as your note taking on the matter is meticulous.
If they do not counter claim, or the Courts find in your favour, the Courts will serve them with a Costs Order which can be enforced by the County Court baillifs. You can also appoint the services of a debt recovery agent. I can recommend a company who can deal with this for you.
If you do intend to claim, claim *only* for the losses incurred. Do not claim for anything else. If you keep the claim very straightforward, it should be easier to complete the paperwork required by the Courts.
I also see no reason why you can not claim for the cost of fitting the correct system without any work being carried out. The current "loss" is the stereo, and you have been quoted for work to repair this. There *will* be consequential loss, and you have been quite reasonable in waiting for them to do the work for you. As I mentioned earlier, look for a solicitor who can offer a small amount of advice for free. As mentioned by other members, you should also run this past the Citizens Advice Bureau.
By the way, is this a small dealer? Are they likely to be so unconcerned that they would just start up a phoenix company? I'd do a Companies House WebCHeck:
Look to see if the company has been dissolved before as this can be quite telling.