Apologies to the barron.
Barristers cannot "cherry pick" cases. We have the "cab rank" rule. This means that as long as it is within their area of expertise and they will be reasonably remunerated they cannot refuse a brief. Thus you cannot say "I don't do publicly funded work".
As for the comment that barristers' hourly rates start at £1,000 per hour, that is the most ludicrous thing I have ever heard. It is well known that it is cheaper to instruct junior counsel rather than solicitors for certain hearings. Certainly as someone less "junior" and with a significant specialism (and having been published and lectured extensively in my field I do have a specialism) my hourly rate is nothing even approaching £240.
My best ever brief fee - and this is less than 10 years ago - involved me prepping a case, travelling for 2 hours each way, waiting in court for 2 hours, and then dealing with a hearing. I received a cheque for ......... wait for it. No actually, you tell me. It hasn't actually won chamber's prize for the lowest ever brief fee but it is close.
Don't get me wrong, I am not pleading poverty, but a grand an hour is just fiction.