I disagree. The court fee would be £205 assuming a claim not exceeding £5000.
https://www.gov.uk/make-court-claim-for-money/court-fees
The process is quick, easy and fairly informal. Just a judge in chambers, usually a desk between you. The main thing, is to present photos, and written evidence, and to be able to explain that they haven't done the job they were contracted to do, that you have had to employ a n other, to sort it out, where they have failed. No special expertise is needed, the judge will expect an ordinary layman to present the case to him. Costs for a solicitor, or an expert, are not normally allowed, in the claim.
Fact - they have left you with a system which is no longer working!
Fact - they are trying to charge you more to fix problems which they appear to have created, through their own incompetence.
Fact - They have tried, and yet failed to fix it, and expect you to pay them more to try other uncertain fixes.
Fact - they have obviously cut corners and ignored regulations!
A Gasguru correctly suggested, TRV's fitted on radiators are now a requirement, rather than an option. I wonder what other corners they might have cut, during the installation? £4300 is an awful lot of money, to just swap out a boiler, and maybe replace 2-port valves. A straight boiler swap, can be done in a day, for £2000.
If there were any obstructed pipes, these should have been evident, when they supposedly flushed the system. I would like to see what the inside of the bottom of the header tank looks like. If the system has been flushed, there should be no gunge or debris in the bottom. Many boiler manufacturers only guarantee their boilers these days - only if the system is thoroughly flushed out, and a filter fitted in the return. Has a filter been fitted?