If £3,000 to get a (detailed) letter sent and a £20,000 - £30,000 bill at the end of if it is settled out of court (and £50k - £100k min if you go to court) then I suggest any of the leading IP firms.
Use the Chamber and Partners guide to the legal profession (link to the London IP firms:
(lhttp://www.chambersandpartners.com/UK/Editorial/34593 ) and find a firm that you think fits. Shop around, get different quotes. Make sure they specialise in contentious matters.
You might see references to soft and hard IP. Copyright is 'soft' IP. You don't want firms that specialise in hard IP alone as that is principally patent-related.
If your budget is not as generous as you thought, find a smaller niche practice (again you can find one in Chambers & Partners) or try some barrister's chambers who might take direct instructions. I have used a junior - mid-level barrister on IP litigation twice and she charges about £150/hr (which is low for a specialist - the larger law firms will charge you £350 to £500/hr). PM me if youl would like her details.
Unless your claim is massive then I suggest you go to a smaller niche firm where you are more likely to deal with a senior lawyer. If you go to a mega-firm then you will be competing for attention with clients such as the large telecoms companies, TV and film studios etc. You might meet a partner at your first meeting but your matter will be dealt with by the most junior of junior lawyers.
Good luck
